Theorem bnj1294 30888

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1294.1	|- ( ph -> A. x e. A ps )
bnj1294.2	|- ( ph -> x e. A )

Assertion

Ref	Expression
bnj1294	|- ( ph -> ps )

Proof of Theorem bnj1294

Step	Hyp	Ref	Expression
1		bnj1294.2	. 2 |- ( ph -> x e. A )
2		bnj1294.1	. 2 |- ( ph -> A. x e. A ps )
3		df-ral 2917	. . 3 |- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
4		sp 2053	. . . 4 |- ( A. x ( x e. A -> ps ) -> ( x e. A -> ps ) )
5	4	impcom 446	. . 3 |- ( ( x e. A /\ A. x ( x e. A -> ps ) ) -> ps )
6	3, 5	sylan2b 492	. 2 |- ( ( x e. A /\ A. x e. A ps ) -> ps )
7	1, 2, 6	syl2anc 693	1 |- ( ph -> ps )

Syntax hints:   -> wi 4   A.wal 1481   e. wcel 1990   A.wral 2912

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-an 386  df-ex 1705  df-ral 2917

This theorem is referenced by:  bnj1379  30901  bnj1121  31053  bnj1279  31086  bnj1286  31087  bnj1296  31089  bnj1421  31110  bnj1450  31118  bnj1489  31124  bnj1501  31135  bnj1523  31139