Theorem r19.29af2 3075

Description: A commonly used pattern based on r19.29 3072. (Contributed by Thierry Arnoux, 17-Dec-2017.) (Proof shortened by OpenAI, 25-Mar-2020.)

Hypotheses

Ref	Expression
r19.29af2.p	|- F/ x ph
r19.29af2.c	|- F/ x ch
r19.29af2.1	|- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
r19.29af2.2	|- ( ph -> E. x e. A ps )

Assertion

Ref	Expression
r19.29af2	|- ( ph -> ch )

Proof of Theorem r19.29af2

Step	Hyp	Ref	Expression
1		r19.29af2.2	. 2 |- ( ph -> E. x e. A ps )
2		r19.29af2.p	. . 3 |- F/ x ph
3		r19.29af2.c	. . 3 |- F/ x ch
4		r19.29af2.1	. . . 4 |- ( ( ( ph /\ x e. A ) /\ ps ) -> ch )
5	4	exp31 630	. . 3 |- ( ph -> ( x e. A -> ( ps -> ch ) ) )
6	2, 3, 5	rexlimd 3026	. 2 |- ( ph -> ( E. x e. A ps -> ch ) )
7	1, 6	mpd 15	1 |- ( ph -> 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:  r19.29af  3076  restmetu  22375  aciunf1lem  29462  fprodex01  29571  locfinreflem  29907  esumrnmpt2  30130  esum2dlem  30154  esum2d  30155  esumiun  30156