Theorem raleqi 2812

Description: Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypothesis

Ref	Expression
raleq1i.1	⊢ A = B

Assertion

Ref	Expression
raleqi	⊢ (∀x ∈ A φ ↔ ∀x ∈ B φ)

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   φ(x)

Proof of Theorem raleqi

Step	Hyp	Ref	Expression
1		raleq1i.1	. 2 ⊢ A = B
2		raleq 2808	. 2 ⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B φ))
3	1, 2	ax-mp 5	1 ⊢ (∀x ∈ A φ ↔ ∀x ∈ B φ)

Syntax hints:   ↔ wb 176   = wceq 1642  ∀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:  ralrab2  3003  ralprg  3776  raltpg  3778  ssofss  4077  ralxp  4826