Theorem bnj228OLD 12518

Description: First-order logic and set theory.

Hypothesis

Ref	Expression
bnj228.1	|- (ph <-> A.x e. A ps)

Assertion

Ref	Expression
bnj228OLD	|- ((x e. A /\ ph) -> ps)

Proof of Theorem bnj228OLD

Step	Hyp	Ref	Expression
1		bnj228.1	. . . 4 |- (ph <-> A.x e. A ps)
2		df-ral 2109	. . . 4 |- (A.x e. A ps <-> A.x(x e. A -> ps))
3	1, 2	bitri 190	. . 3 |- (ph <-> A.x(x e. A -> ps))
4		ax-4 1319	. . 3 |- (A.x(x e. A -> ps) -> (x e. A -> ps))
5	3, 4	sylbi 216	. 2 |- (ph -> (x e. A -> ps))
6	5	impcom 378	1 |- ((x e. A /\ ph) -> ps)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   e. wcel 1300  A.wral 2105

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 1319

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

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