Theorem r19.21vOLD 2179

Description: Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers.

Assertion

Ref	Expression
r19.21vOLD	|- (A.x e. A (ph -> ps) <-> (ph -> A.x e. A ps))

Distinct variable group:   ph,x

Proof of Theorem r19.21vOLD

Step	Hyp	Ref	Expression
1		ax-17 1317	. . 3 |- (ph -> A.xph)
2	1	ax-gen 1305	. 2 |- A.x(ph -> A.xph)
3		r19.21t 2177	. 2 |- (A.x(ph -> A.xph) -> (A.x e. A (ph -> ps) <-> (ph -> A.x e. A ps)))
4	2, 3	ax-mp 7	1 |- (A.x e. A (ph -> ps) <-> (ph -> A.x e. A ps))

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296  A.wral 2105

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324

This theorem depends on definitions:  df-bi 164  df-an 242  df-ral 2109

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