Theorem r19.29af 2497

Description: A commonly used pattern based on r19.29 2494 (Contributed by Thierry Arnoux, 29-Nov-2017.)

Hypotheses

Ref	Expression
r19.29af.0	|- F/ x ph
r19.29af.1	|- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
r19.29af.2	|- ( ph -> E. x e. A ps )

Assertion

Ref	Expression
r19.29af	|- ( ph -> ch )

Distinct variable group:   ch, x

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

Proof of Theorem r19.29af

Step	Hyp	Ref	Expression
1		r19.29af.0	. 2 |- F/ x ph
2		nfv 1461	. 2 |- F/ x ch
3		r19.29af.1	. 2 |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
4		r19.29af.2	. 2 |- ( ph -> E. x e. A ps )
5	1, 2, 3, 4	r19.29af2 2496	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   /\ wa 102   F/wnf 1389   e. wcel 1433   E.wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-ral 2353  df-rex 2354

This theorem is referenced by:  r19.29a  2498  supinfneg  8683  infsupneg  8684