Theorem 2spim 15412

Description: Double substitution, as in spim 1752. (Contributed by BJ, 17-Oct-2019.)

Hypotheses

Ref	Expression
2spim.nfx	|- F/ x ch
2spim.nfz	|- F/ z ch
2spim.1	|- ( ( x = y /\ z = t ) -> ( ps -> ch ) )

Assertion

Ref	Expression
2spim	|- ( A. z A. x ps -> ch )

Distinct variable groups:   x, z   x, t

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

Proof of Theorem 2spim

Step	Hyp	Ref	Expression
1		2spim.nfz	. 2 |- F/ z ch
2		2spim.nfx	. . . 4 |- F/ x ch
3	2	a1i 9	. . 3 |- ( z = t -> F/ x ch )
4		2spim.1	. . . . 5 |- ( ( x = y /\ z = t ) -> ( ps -> ch ) )
5	4	expcom 116	. . . 4 |- ( z = t -> ( x = y -> ( ps -> ch ) ) )
6	5	alrimiv 1888	. . 3 |- ( z = t -> A. x ( x = y -> ( ps -> ch ) ) )
7	3, 6	spimd 15411	. 2 |- ( z = t -> ( A. x ps -> ch ) )
8	1, 7	spim 1752	1 |- ( A. z A. x ps -> ch )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   F/wnf 1474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-nf 1475

This theorem is referenced by:  ch2var  15413