Theorem bnj1509 13169

Description: First-order logic and set theory.

Hypotheses

Ref	Expression
bnj1509.1	|- A = B
bnj1509.2	|- (ph -> A.x e. B ps)

Assertion

Ref	Expression
bnj1509	|- (ph -> A.x e. A ps)

Proof of Theorem bnj1509

Step	Hyp	Ref	Expression
1		bnj1509.2	. 2 |- (ph -> A.x e. B ps)
2		bnj1509.1	. . . . 5 |- A = B
3	2	eleq2i 1961	. . . 4 |- (x e. A <-> x e. B)
4	3	imbi1i 203	. . 3 |- ((x e. A -> ps) <-> (x e. B -> ps))
5	4	ralbii2 2131	. 2 |- (A.x e. A ps <-> A.x e. B ps)
6	1, 5	sylibr 217	1 |- (ph -> A.x e. A ps)

Syntax hints:   -> wi 3   = wceq 1298   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-gen 1305  ax-17 1317  ax-4 1319  ax-5o 1321  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-cleq 1877  df-clel 1880  df-ral 2109

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