Theorem pm14.123b 38627

Description: Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)

Assertion

Ref	Expression
pm14.123b	|- ( ( A e. V /\ B e. W ) -> ( ( A. z A. w ( ph -> ( z = A /\ w = B ) ) /\ [. A / z ]. [. B / w ]. ph ) <-> ( A. z A. w ( ph -> ( z = A /\ w = B ) ) /\ E. z E. w ph ) ) )

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

Allowed substitution hints:   ph( z, w)   V( z, w)   W( z, w)

Proof of Theorem pm14.123b

Step	Hyp	Ref	Expression
1		2sbc5g 38617	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( E. z E. w ( ( z = A /\ w = B ) /\ ph ) <-> [. A / z ]. [. B / w ]. ph ) )
2	1	adantr 481	. . 3 |- ( ( ( A e. V /\ B e. W ) /\ A. z A. w ( ph -> ( z = A /\ w = B ) ) ) -> ( E. z E. w ( ( z = A /\ w = B ) /\ ph ) <-> [. A / z ]. [. B / w ]. ph ) )
3		nfa1 2028	. . . . 5 |- F/ z A. z A. w ( ph -> ( z = A /\ w = B ) )
4		nfa2 2040	. . . . . 6 |- F/ w A. z A. w ( ph -> ( z = A /\ w = B ) )
5		simpr 477	. . . . . . 7 |- ( ( ( z = A /\ w = B ) /\ ph ) -> ph )
6		2sp 2056	. . . . . . . 8 |- ( A. z A. w ( ph -> ( z = A /\ w = B ) ) -> ( ph -> ( z = A /\ w = B ) ) )
7	6	ancrd 577	. . . . . . 7 |- ( A. z A. w ( ph -> ( z = A /\ w = B ) ) -> ( ph -> ( ( z = A /\ w = B ) /\ ph ) ) )
8	5, 7	impbid2 216	. . . . . 6 |- ( A. z A. w ( ph -> ( z = A /\ w = B ) ) -> ( ( ( z = A /\ w = B ) /\ ph ) <-> ph ) )
9	4, 8	exbid 2091	. . . . 5 |- ( A. z A. w ( ph -> ( z = A /\ w = B ) ) -> ( E. w ( ( z = A /\ w = B ) /\ ph ) <-> E. w ph ) )
10	3, 9	exbid 2091	. . . 4 |- ( A. z A. w ( ph -> ( z = A /\ w = B ) ) -> ( E. z E. w ( ( z = A /\ w = B ) /\ ph ) <-> E. z E. w ph ) )
11	10	adantl 482	. . 3 |- ( ( ( A e. V /\ B e. W ) /\ A. z A. w ( ph -> ( z = A /\ w = B ) ) ) -> ( E. z E. w ( ( z = A /\ w = B ) /\ ph ) <-> E. z E. w ph ) )
12	2, 11	bitr3d 270	. 2 |- ( ( ( A e. V /\ B e. W ) /\ A. z A. w ( ph -> ( z = A /\ w = B ) ) ) -> ( [. A / z ]. [. B / w ]. ph <-> E. z E. w ph ) )
13	12	pm5.32da 673	1 |- ( ( A e. V /\ B e. W ) -> ( ( A. z A. w ( ph -> ( z = A /\ w = B ) ) /\ [. A / z ]. [. B / w ]. ph ) <-> ( A. z A. w ( ph -> ( z = A /\ w = B ) ) /\ E. z E. w ph ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   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:  pm14.123c  38628