Theorem raleqbidv 2820

Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)

Hypotheses

Ref	Expression
raleqbidv.1	|- (ph -> A = B)
raleqbidv.2	|- (ph -> (ps <-> ch))

Assertion

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

Distinct variable groups:   x,A   x,B   ph,x

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

Proof of Theorem raleqbidv

Step	Hyp	Ref	Expression
1		raleqbidv.1	. . 3 |- (ph -> A = B)
2	1	raleqdv 2814	. 2 |- (ph -> (A.x e. A ps <-> A.x e. B ps))
3		raleqbidv.2	. . 3 |- (ph -> (ps <-> ch))
4	3	ralbidv 2635	. 2 |- (ph -> (A.x e. B ps <-> A.x e. B ch))
5	2, 4	bitrd 244	1 |- (ph -> (A.x e. A ps <-> A.x e. B ch))

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642  A.wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620

This theorem is referenced by:  fmpt2x  5731