Theorem subtr2 32309

Description: Transitivity of implicit substitution into a wff. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

Hypotheses

Ref	Expression
subtr.1	|- F/_ x A
subtr.2	|- F/_ x B
subtr2.3	|- F/ x ps
subtr2.4	|- F/ x ch
subtr2.5	|- ( x = A -> ( ph <-> ps ) )
subtr2.6	|- ( x = B -> ( ph <-> ch ) )

Assertion

Ref	Expression
subtr2	|- ( ( A e. C /\ B e. D ) -> ( A = B -> ( ps <-> ch ) ) )

Proof of Theorem subtr2

Step	Hyp	Ref	Expression
1		subtr.1	. . 3 |- F/_ x A
2		subtr.2	. . . . 5 |- F/_ x B
3	1, 2	nfeq 2776	. . . 4 |- F/ x A = B
4		subtr2.3	. . . . 5 |- F/ x ps
5		subtr2.4	. . . . 5 |- F/ x ch
6	4, 5	nfbi 1833	. . . 4 |- F/ x ( ps <-> ch )
7	3, 6	nfim 1825	. . 3 |- F/ x ( A = B -> ( ps <-> ch ) )
8		eqeq1 2626	. . . 4 |- ( x = A -> ( x = B <-> A = B ) )
9		subtr2.5	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
10	9	bibi1d 333	. . . 4 |- ( x = A -> ( ( ph <-> ch ) <-> ( ps <-> ch ) ) )
11	8, 10	imbi12d 334	. . 3 |- ( x = A -> ( ( x = B -> ( ph <-> ch ) ) <-> ( A = B -> ( ps <-> ch ) ) ) )
12		subtr2.6	. . 3 |- ( x = B -> ( ph <-> ch ) )
13	1, 7, 11, 12	vtoclgf 3264	. 2 |- ( A e. C -> ( A = B -> ( ps <-> ch ) ) )
14	13	adantr 481	1 |- ( ( A e. C /\ B e. D ) -> ( A = B -> ( ps <-> ch ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751

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

This theorem is referenced by: (None)