Theorem 2sbc6g 38616

Description: Theorem *13.21 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)

Assertion

Ref	Expression
2sbc6g	|- ( ( A e. C /\ B e. D ) -> ( A. z A. w ( ( z = A /\ w = B ) -> ph ) <-> [. A / z ]. [. B / w ]. ph ) )

Distinct variable groups:   z, w, A   w, B, z

Allowed substitution hints:   ph( z, w)   C( z, w)   D( z, w)

Proof of Theorem 2sbc6g

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2633	. . . . . . 7 |- ( y = B -> ( w = y <-> w = B ) )
2	1	anbi2d 740	. . . . . 6 |- ( y = B -> ( ( z = x /\ w = y ) <-> ( z = x /\ w = B ) ) )
3	2	imbi1d 331	. . . . 5 |- ( y = B -> ( ( ( z = x /\ w = y ) -> ph ) <-> ( ( z = x /\ w = B ) -> ph ) ) )
4	3	2albidv 1851	. . . 4 |- ( y = B -> ( A. z A. w ( ( z = x /\ w = y ) -> ph ) <-> A. z A. w ( ( z = x /\ w = B ) -> ph ) ) )
5		dfsbcq 3437	. . . . 5 |- ( y = B -> ( [. y / w ]. ph <-> [. B / w ]. ph ) )
6	5	sbcbidv 3490	. . . 4 |- ( y = B -> ( [. x / z ]. [. y / w ]. ph <-> [. x / z ]. [. B / w ]. ph ) )
7	4, 6	bibi12d 335	. . 3 |- ( y = B -> ( ( A. z A. w ( ( z = x /\ w = y ) -> ph ) <-> [. x / z ]. [. y / w ]. ph ) <-> ( A. z A. w ( ( z = x /\ w = B ) -> ph ) <-> [. x / z ]. [. B / w ]. ph ) ) )
8		eqeq2 2633	. . . . . . 7 |- ( x = A -> ( z = x <-> z = A ) )
9	8	anbi1d 741	. . . . . 6 |- ( x = A -> ( ( z = x /\ w = B ) <-> ( z = A /\ w = B ) ) )
10	9	imbi1d 331	. . . . 5 |- ( x = A -> ( ( ( z = x /\ w = B ) -> ph ) <-> ( ( z = A /\ w = B ) -> ph ) ) )
11	10	2albidv 1851	. . . 4 |- ( x = A -> ( A. z A. w ( ( z = x /\ w = B ) -> ph ) <-> A. z A. w ( ( z = A /\ w = B ) -> ph ) ) )
12		dfsbcq 3437	. . . 4 |- ( x = A -> ( [. x / z ]. [. B / w ]. ph <-> [. A / z ]. [. B / w ]. ph ) )
13	11, 12	bibi12d 335	. . 3 |- ( x = A -> ( ( A. z A. w ( ( z = x /\ w = B ) -> ph ) <-> [. x / z ]. [. B / w ]. ph ) <-> ( A. z A. w ( ( z = A /\ w = B ) -> ph ) <-> [. A / z ]. [. B / w ]. ph ) ) )
14		vex 3203	. . . . 5 |- x e. _V
15	14	sbc6 3462	. . . 4 |- ( [. x / z ]. [. y / w ]. ph <-> A. z ( z = x -> [. y / w ]. ph ) )
16		19.21v 1868	. . . . . 6 |- ( A. w ( z = x -> ( w = y -> ph ) ) <-> ( z = x -> A. w ( w = y -> ph ) ) )
17		impexp 462	. . . . . . 7 |- ( ( ( z = x /\ w = y ) -> ph ) <-> ( z = x -> ( w = y -> ph ) ) )
18	17	albii 1747	. . . . . 6 |- ( A. w ( ( z = x /\ w = y ) -> ph ) <-> A. w ( z = x -> ( w = y -> ph ) ) )
19		vex 3203	. . . . . . . 8 |- y e. _V
20	19	sbc6 3462	. . . . . . 7 |- ( [. y / w ]. ph <-> A. w ( w = y -> ph ) )
21	20	imbi2i 326	. . . . . 6 |- ( ( z = x -> [. y / w ]. ph ) <-> ( z = x -> A. w ( w = y -> ph ) ) )
22	16, 18, 21	3bitr4ri 293	. . . . 5 |- ( ( z = x -> [. y / w ]. ph ) <-> A. w ( ( z = x /\ w = y ) -> ph ) )
23	22	albii 1747	. . . 4 |- ( A. z ( z = x -> [. y / w ]. ph ) <-> A. z A. w ( ( z = x /\ w = y ) -> ph ) )
24	15, 23	bitr2i 265	. . 3 |- ( A. z A. w ( ( z = x /\ w = y ) -> ph ) <-> [. x / z ]. [. y / w ]. ph )
25	7, 13, 24	vtocl2g 3270	. 2 |- ( ( B e. D /\ A e. C ) -> ( A. z A. w ( ( z = A /\ w = B ) -> ph ) <-> [. A / z ]. [. B / w ]. ph ) )
26	25	ancoms 469	1 |- ( ( A e. C /\ B e. D ) -> ( A. z A. w ( ( z = A /\ w = B ) -> ph ) <-> [. A / z ]. [. B / w ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436

This theorem is referenced by:  pm14.123a  38626