Theorem 2spim 9241

Description: Double substitution, as in spim 1623. (Contributed by BJ, 17-Oct-2019.)

Hypotheses

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

Assertion

Ref	Expression
2spim	|- (A.zA.xps -> 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/zch
2		2spim.nfx	. . . 4 |- F/xch
3	2	a1i 9	. . 3 |- (z = t -> F/xch)
4		2spim.1	. . . . 5 |- ((x = y /\ z = t) -> (ps -> ch))
5	4	expcom 109	. . . 4 |- (z = t -> (x = y -> (ps -> ch)))
6	5	alrimiv 1751	. . 3 |- (z = t -> A.x(x = y -> (ps -> ch)))
7	3, 6	spimd 9240	. 2 |- (z = t -> (A.xps -> ch))
8	1, 7	spim 1623	1 |- (A.zA.xps -> ch)

Syntax hints:   -> wi 4   /\ wa 97  A.wal 1240   F/wnf 1346

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-nf 1347

This theorem is referenced by:  ch2var  9242