Theorem birala 1228

Description: Inference adding restricted universal quantifier to both sides of an equivalence.

Hypothesis

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

Assertion

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

Proof of Theorem birala

Step	Hyp	Ref	Expression
1		birala.1	. . . 4 |- (x e. A -> (ph <-> ps))
2	1	pm5.74i 443	. . 3 |- ((x e. A -> ph) <-> (x e. A -> ps))
3	2	bial 695	. 2 |- (A.x(x e. A -> ph) <-> A.x(x e. A -> ps))
4		df-ral 1205	. 2 |- (A.x e. A ph <-> A.x(x e. A -> ph))
5		df-ral 1205	. 2 |- (A.x e. A ps <-> A.x(x e. A -> ps))
6	3, 4, 5	3bitr4 158	1 |- (A.x e. A ph <-> A.x e. A ps)

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672   e. wcel 1092  A.wral 1201

This theorem is referenced by:  fvreseq 2882  aceq4 3557  hods 5606  large 5700  elat2 5739

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

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