Theorem spcdvw 42426

Description: A version of spcdv 3291 where ps and ch are direct substitutions of each other. This theorem is useful because it does not require ph and x to be distinct variables. (Contributed by Emmett Weisz, 12-Apr-2020.)

Hypotheses

Ref	Expression
spcdvw.1	|- ( ph -> A e. B )
spcdvw.2	|- ( x = A -> ( ps <-> ch ) )

Assertion

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

Distinct variable groups:   x, A   ch, x

Allowed substitution hints:   ph( x)   ps( x)   B( x)

Proof of Theorem spcdvw

Step	Hyp	Ref	Expression
1		spcdvw.2	. . . 4 |- ( x = A -> ( ps <-> ch ) )
2	1	biimpd 219	. . 3 |- ( x = A -> ( ps -> ch ) )
3	2	ax-gen 1722	. 2 |- A. x ( x = A -> ( ps -> ch ) )
4		spcdvw.1	. 2 |- ( ph -> A e. B )
5		nfv 1843	. . 3 |- F/ x ch
6		nfcv 2764	. . 3 |- F/_ x A
7	5, 6	spcimgft 3284	. 2 |- ( A. x ( x = A -> ( ps -> ch ) ) -> ( A e. B -> ( A. x ps -> ch ) ) )
8	3, 4, 7	mpsyl 68	1 |- ( ph -> ( A. x ps -> ch ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990

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:  setrec1lem4  42437