Theorem r19.45av 1766

Description: Restricted version of one direction of Theorem 19.45 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.)

Assertion

Ref	Expression
r19.45av	|- (E.x e. A (ph \/ ps) -> (ph \/ E.x e. A ps))

Distinct variable group:   ph,x

Proof of Theorem r19.45av

Step	Hyp	Ref	Expression
1		r19.43 1764	. 2 |- (E.x e. A (ph \/ ps) <-> (E.x e. A ph \/ E.x e. A ps))
2		idd 61	. . . 4 |- (x e. A -> (ph -> ph))
3	2	r19.23aiv 1742	. . 3 |- (E.x e. A ph -> ph)
4	3	orim1i 337	. 2 |- ((E.x e. A ph \/ E.x e. A ps) -> (ph \/ E.x e. A ps))
5	1, 4	sylbi 199	1 |- (E.x e. A (ph \/ ps) -> (ph \/ E.x e. A ps))

Syntax hints:   -> wi 3   \/ wo 222   e. wcel 957  E.wrex 1645

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 962  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-rex 1649

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