Theorem biraldv2 1221

Description: Formula-building rule for restricted universal quantifier (deduction rule).

Hypothesis

Ref	Expression
biraldv2.1	|- (ph -> ((x e. A -> ps) <-> (x e. B -> ch)))

Assertion

Ref	Expression
biraldv2	|- (ph -> (A.x e. A ps <-> A.x e. B ch))

Distinct variable group(s):   ph,x

Proof of Theorem biraldv2

Step	Hyp	Ref	Expression
1		biraldv2.1	. . 3 |- (ph -> ((x e. A -> ps) <-> (x e. B -> ch)))
2	1	bialdv 935	. 2 |- (ph -> (A.x(x e. A -> ps) <-> A.x(x e. B -> ch)))
3		df-ral 1205	. 2 |- (A.x e. A ps <-> A.x(x e. A -> ps))
4		df-ral 1205	. 2 |- (A.x e. B ch <-> A.x(x e. B -> ch))
5	2, 3, 4	3bitr4g 428	1 |- (ph -> (A.x e. A ps <-> A.x e. B ch))

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

This theorem is referenced by:  oneqmini 2272  zornlem1 3603  iscard2 3660

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  ax-17 925

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

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