Theorem r19.37 2761

Description: Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypothesis

Ref	Expression
r19.37.1	|- F/xph

Assertion

Ref	Expression
r19.37	|- (E.x e. A (ph -> ps) -> (ph -> E.x e. A ps))

Proof of Theorem r19.37

Step	Hyp	Ref	Expression
1		r19.35 2759	. 2 |- (E.x e. A (ph -> ps) <-> (A.x e. A ph -> E.x e. A ps))
2		r19.37.1	. . . 4 |- F/xph
3		ax-1 6	. . . 4 |- (ph -> (x e. A -> ph))
4	2, 3	ralrimi 2696	. . 3 |- (ph -> A.x e. A ph)
5	4	imim1i 54	. 2 |- ((A.x e. A ph -> E.x e. A ps) -> (ph -> E.x e. A ps))
6	1, 5	sylbi 187	1 |- (E.x e. A (ph -> ps) -> (ph -> E.x e. A ps))

Syntax hints:   -> wi 4   F/wnf 1544   e. wcel 1710  A.wral 2615  E.wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2620  df-rex 2621

This theorem is referenced by:  r19.37av  2762