Theorem rexlimd3 39335

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

Hypotheses

Ref	Expression
rexlimd3.1	|- F/ x ph
rexlimd3.2	|- F/ x ch
rexlimd3.3	|- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )

Assertion

Ref	Expression
rexlimd3	|- ( ph -> ( E. x e. A ps -> ch ) )

Proof of Theorem rexlimd3

Step	Hyp	Ref	Expression
1		rexlimd3.1	. 2 |- F/ x ph
2		rexlimd3.2	. 2 |- F/ x ch
3		rexlimd3.3	. . 3 |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
4	3	exp31 630	. 2 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
5	1, 2, 4	rexlimd 3026	1 |- ( ph -> ( E. x e. A ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 384   F/wnf 1708   e. wcel 1990   E.wrex 2913

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-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-ral 2917  df-rex 2918

This theorem is referenced by: (None)