Theorem ralbida 1649

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

Hypotheses

Ref	Expression
ralbida.1	|- (ph -> A.xph)
ralbida.2	|- ((ph /\ x e. A) -> (ps <-> ch))

Assertion

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

Proof of Theorem ralbida

Step	Hyp	Ref	Expression
1		ralbida.1	. . 3 |- (ph -> A.xph)
2		ralbida.2	. . . 4 |- ((ph /\ x e. A) -> (ps <-> ch))
3	2	pm5.74da 584	. . 3 |- (ph -> ((x e. A -> ps) <-> (x e. A -> ch)))
4	1, 3	albid 1100	. 2 |- (ph -> (A.x(x e. A -> ps) <-> A.x(x e. A -> ch)))
5		df-ral 1641	. 2 |- (A.x e. A ps <-> A.x(x e. A -> ps))
6		df-ral 1641	. 2 |- (A.x e. A ch <-> A.x(x e. A -> ch))
7	4, 5, 6	3bitr4g 553	1 |- (ph -> (A.x e. A ps <-> A.x e. A ch))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   e. wcel 955  A.wral 1637

This theorem is referenced by:  ralbidva 1651  ralbid 1653  2ralbida 1669  r19.15 1745  iineq2 2569  mapxpen 4475  xpmapenlem5 4480  clm0 7021  clm0nns 7023

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-4 970  ax-5o 972

This theorem depends on definitions:  df-bi 147  df-an 225  df-ral 1641

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