Theorem r19.29af 2649

Description: A commonly used pattern based on r19.29 2645. (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 1552	. 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 2648	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   /\ wa 104   F/wnf 1484   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-tru 1376  df-nf 1485  df-ral 2491  df-rex 2492

This theorem is referenced by:  r19.29an  2650  r19.29a  2651  suplocsrlem  7956  supinfneg  9751  infsupneg  9752  pw1nct  16142