Theorem rspcedvdw 2930

Description: Version of rspcedvd 2929 where the implicit substitution hypothesis does not have an antecedent, which also avoids a disjoint variable condition on ph , x. (Contributed by SN, 20-Aug-2024.)

Hypotheses

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

Assertion

Ref	Expression
rspcedvdw	|- ( ph -> E. x e. B ps )

Distinct variable groups:   x, A   x, B   ch, x

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

Proof of Theorem rspcedvdw

Step	Hyp	Ref	Expression
1		rspcedvdw.1	. 2 |- ( ph -> A e. B )
2		rspcedvdw.2	. 2 |- ( ph -> ch )
3		rspcedvdw.s	. . 3 |- ( x = A -> ( ps <-> ch ) )
4	3	rspcev 2923	. 2 |- ( ( A e. B /\ ch ) -> E. x e. B ps )
5	1, 2, 4	syl2anc 411	1 |- ( ph -> E. x e. B ps )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2205   E.wrex 2523

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817

This theorem is referenced by:  ballotfilem1c  13195  ballotfilemrc  13218