Theorem ch2var 10578

Description: Implicit substitution of y for x and t for z into a theorem. (Contributed by BJ, 17-Oct-2019.)

Hypotheses

Ref	Expression
ch2var.nfx	|- F/ x ps
ch2var.nfz	|- F/ z ps
ch2var.maj	|- ( ( x = y /\ z = t ) -> ( ph <-> ps ) )
ch2var.min	|- ph

Assertion

Ref	Expression
ch2var	|- ps

Distinct variable groups:   x, z   x, t

Allowed substitution hints:   ph( x, y, z, t)   ps( x, y, z, t)

Proof of Theorem ch2var

Step	Hyp	Ref	Expression
1		ch2var.nfx	. . 3 |- F/ x ps
2		ch2var.nfz	. . 3 |- F/ z ps
3		ch2var.maj	. . . 4 |- ( ( x = y /\ z = t ) -> ( ph <-> ps ) )
4	3	biimpd 142	. . 3 |- ( ( x = y /\ z = t ) -> ( ph -> ps ) )
5	1, 2, 4	2spim 10577	. 2 |- ( A. z A. x ph -> ps )
6		ch2var.min	. . 3 |- ph
7	6	ax-gen 1378	. 2 |- A. x ph
8	5, 7	mpg 1380	1 |- ps

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   F/wnf 1389

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-nf 1390

This theorem is referenced by:  ch2varv  10579