Theorem ralbidv2 2552

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Apr-1997.)

Hypothesis

Ref	Expression
ralbidv2.1	|- ( ph -> ( ( x e. A -> ps ) <-> ( x e. B -> ch ) ) )

Assertion

Ref	Expression
ralbidv2	|- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   ch( x)   A( x)   B( x)

Proof of Theorem ralbidv2

Step	Hyp	Ref	Expression
1		ralbidv2.1	. . 3 |- ( ph -> ( ( x e. A -> ps ) <-> ( x e. B -> ch ) ) )
2	1	albidv 1877	. 2 |- ( ph -> ( A. x ( x e. A -> ps ) <-> A. x ( x e. B -> ch ) ) )
3		df-ral 2533	. 2 |- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
4		df-ral 2533	. 2 |- ( A. x e. B ch <-> A. x ( x e. B -> ch ) )
5	2, 3, 4	3bitr4g 223	1 |- ( ph -> ( A. x e. A ps <-> A. x e. B ch ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   A.wal 1400   e. wcel 2209   A.wral 2528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-17 1579

This proof depends on definitions:  df-bi 117  df-ral 2533

This theorem is used by:  ralss  3314  dfsmo2  6558  raluz  9978  isprm3  12896  metcnp  15613  sscoll2  17014