Theorem raleqbidv 2820

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

Hypotheses

Ref	Expression
raleqbidv.1	⊢ (φ → A = B)
raleqbidv.2	⊢ (φ → (ψ ↔ χ))

Assertion

Ref	Expression
raleqbidv	⊢ (φ → (∀x ∈ A ψ ↔ ∀x ∈ B χ))

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

Allowed substitution hints:   ψ(x)   χ(x)

Proof of Theorem raleqbidv

Step	Hyp	Ref	Expression
1		raleqbidv.1	. . 3 ⊢ (φ → A = B)
2	1	raleqdv 2814	. 2 ⊢ (φ → (∀x ∈ A ψ ↔ ∀x ∈ B ψ))
3		raleqbidv.2	. . 3 ⊢ (φ → (ψ ↔ χ))
4	3	ralbidv 2635	. 2 ⊢ (φ → (∀x ∈ B ψ ↔ ∀x ∈ B χ))
5	2, 4	bitrd 244	1 ⊢ (φ → (∀x ∈ A ψ ↔ ∀x ∈ B χ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   = wceq 1642  ∀wral 2615

This proof depends on 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 proof 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 used by:  fmpt2x  5731