Theorem pm14.122b 38624

Description: Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)

Assertion

Ref	Expression
pm14.122b	|- ( A e. V -> ( ( A. x ( ph -> x = A ) /\ [. A / x ]. ph ) <-> ( A. x ( ph -> x = A ) /\ E. x ph ) ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem pm14.122b

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2633	. . . . . 6 |- ( y = A -> ( x = y <-> x = A ) )
2	1	imbi2d 330	. . . . 5 |- ( y = A -> ( ( ph -> x = y ) <-> ( ph -> x = A ) ) )
3	2	albidv 1849	. . . 4 |- ( y = A -> ( A. x ( ph -> x = y ) <-> A. x ( ph -> x = A ) ) )
4		dfsbcq 3437	. . . . 5 |- ( y = A -> ( [. y / x ]. ph <-> [. A / x ]. ph ) )
5	4	bibi1d 333	. . . 4 |- ( y = A -> ( ( [. y / x ]. ph <-> E. x ph ) <-> ( [. A / x ]. ph <-> E. x ph ) ) )
6	3, 5	imbi12d 334	. . 3 |- ( y = A -> ( ( A. x ( ph -> x = y ) -> ( [. y / x ]. ph <-> E. x ph ) ) <-> ( A. x ( ph -> x = A ) -> ( [. A / x ]. ph <-> E. x ph ) ) ) )
7		sbc5 3460	. . . 4 |- ( [. y / x ]. ph <-> E. x ( x = y /\ ph ) )
8		nfa1 2028	. . . . 5 |- F/ x A. x ( ph -> x = y )
9		simpr 477	. . . . . 6 |- ( ( x = y /\ ph ) -> ph )
10		ancr 572	. . . . . . 7 |- ( ( ph -> x = y ) -> ( ph -> ( x = y /\ ph ) ) )
11	10	sps 2055	. . . . . 6 |- ( A. x ( ph -> x = y ) -> ( ph -> ( x = y /\ ph ) ) )
12	9, 11	impbid2 216	. . . . 5 |- ( A. x ( ph -> x = y ) -> ( ( x = y /\ ph ) <-> ph ) )
13	8, 12	exbid 2091	. . . 4 |- ( A. x ( ph -> x = y ) -> ( E. x ( x = y /\ ph ) <-> E. x ph ) )
14	7, 13	syl5bb 272	. . 3 |- ( A. x ( ph -> x = y ) -> ( [. y / x ]. ph <-> E. x ph ) )
15	6, 14	vtoclg 3266	. 2 |- ( A e. V -> ( A. x ( ph -> x = A ) -> ( [. A / x ]. ph <-> E. x ph ) ) )
16	15	pm5.32d 671	1 |- ( A e. V -> ( ( A. x ( ph -> x = A ) /\ [. A / x ]. ph ) <-> ( A. x ( ph -> x = A ) /\ E. x 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-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-v 3202  df-sbc 3436

This theorem is referenced by:  pm14.122c  38625