Theorem spimd 12972

Description: Deduction form of spim 1716. (Contributed by BJ, 17-Oct-2019.)

Hypotheses

Ref	Expression
spimd.nf	|- ( ph -> F/ x ch )
spimd.1	|- ( ph -> A. x ( x = y -> ( ps -> ch ) ) )

Assertion

Ref	Expression
spimd	|- ( ph -> ( A. x ps -> ch ) )

Proof of Theorem spimd

Step	Hyp	Ref	Expression
1		spimd.nf	. 2 |- ( ph -> F/ x ch )
2		spimd.1	. 2 |- ( ph -> A. x ( x = y -> ( ps -> ch ) ) )
3		spimt 1714	. 2 |- ( ( F/ x ch /\ A. x ( x = y -> ( ps -> ch ) ) ) -> ( A. x ps -> ch ) )
4	1, 2, 3	syl2anc 408	1 |- ( ph -> ( A. x ps -> ch ) )

Syntax hints:   -> wi 4   A.wal 1329   F/wnf 1436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-nf 1437

This theorem is referenced by:  2spim  12973