Theorem r19.29a 2613

Description: A commonly used pattern based on r19.29 2607. (Contributed by Thierry Arnoux, 22-Nov-2017.)

Hypotheses

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

Assertion

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

Distinct variable groups:   ch, x   ph, x

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

Proof of Theorem r19.29a

Step	Hyp	Ref	Expression
1		nfv 1521	. 2 |- F/ x ph
2		r19.29a.1	. 2 |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
3		r19.29a.2	. 2 |- ( ph -> E. x e. A ps )
4	1, 2, 3	r19.29af 2611	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   E.wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-ral 2453  df-rex 2454

This theorem is referenced by:  cnegexlem3  8096  cnegex  8097  modqmuladdnn0  10324  uzwodc  11992  1arith  12319  neitx  13062