Theorem raleqdv 2814

Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)

Hypothesis

Ref	Expression
raleq1d.1	|- (ph -> A = B)

Assertion

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

Distinct variable groups:   x,A   x,B

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

Proof of Theorem raleqdv

Step	Hyp	Ref	Expression
1		raleq1d.1	. 2 |- (ph -> A = B)
2		raleq 2808	. 2 |- (A = B -> (A.x e. A ps <-> A.x e. B ps))
3	1, 2	syl 15	1 |- (ph -> (A.x e. A ps <-> A.x e. B ps))

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:  raleqbidv  2820  raleqbidva  2822