Theorem rexlimdva2 2628

Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypothesis

Ref	Expression
rexlimdva2.1	|- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )

Assertion

Ref	Expression
rexlimdva2	|- ( ph -> ( E. x e. A ps -> ch ) )

Distinct variable groups:   ch, x   ph, x

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

Proof of Theorem rexlimdva2

Step	Hyp	Ref	Expression
1		rexlimdva2.1	. . 3 |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
2	1	exp31 364	. 2 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
3	2	rexlimdv 2624	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2178   E.wrex 2487

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-ral 2491  df-rex 2492

This theorem is referenced by:  ctssdclemn0  7238  ctssdc  7241  suplocexprlemru  7867  suplocexprlemloc  7869  suplocsrlemb  7954  aptap  8758  4sqlemffi  12834  4sqleminfi  12835  4sqexercise2  12837  4sqlemsdc  12838  ennnfonelemhom  12901  gsumfzval  13338  reldvdsrsrg  13969  innei  14750  ivthinclemlr  15224  ivthinclemur  15226  limccnpcntop  15262  limccoap  15265  2lgslem1c  15682  2lgslem3a1  15689  2lgslem3b1  15690  2lgslem3c1  15691  2lgslem3d1  15692