Theorem subtr 32308

Description: Transitivity of implicit substitution. (Contributed by Jeff Hankins, 13-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

Hypotheses

Ref	Expression
subtr.1	|- F/_ x A
subtr.2	|- F/_ x B
subtr.3	|- F/_ x Y
subtr.4	|- F/_ x Z
subtr.5	|- ( x = A -> X = Y )
subtr.6	|- ( x = B -> X = Z )

Assertion

Ref	Expression
subtr	|- ( ( A e. C /\ B e. D ) -> ( A = B -> Y = Z ) )

Proof of Theorem subtr

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		subtr.3	. . . . 5 |- F/_ x Y
5		subtr.4	. . . . 5 |- F/_ x Z
6	4, 5	nfeq 2776	. . . 4 |- F/ x Y = Z
7	3, 6	nfim 1825	. . 3 |- F/ x ( A = B -> Y = Z )
8		eqeq1 2626	. . . 4 |- ( x = A -> ( x = B <-> A = B ) )
9		subtr.5	. . . . 5 |- ( x = A -> X = Y )
10	9	eqeq1d 2624	. . . 4 |- ( x = A -> ( X = Z <-> Y = Z ) )
11	8, 10	imbi12d 334	. . 3 |- ( x = A -> ( ( x = B -> X = Z ) <-> ( A = B -> Y = Z ) ) )
12		subtr.6	. . 3 |- ( x = B -> X = Z )
13	1, 7, 11, 12	vtoclgf 3264	. 2 |- ( A e. C -> ( A = B -> Y = Z ) )
14	13	adantr 481	1 |- ( ( A e. C /\ B e. D ) -> ( A = B -> Y = Z ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   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)