Theorem raleqbi1dv 2816

Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)

Hypothesis

Ref	Expression
raleqd.1	⊢ (A = B → (φ ↔ ψ))

Assertion

Ref	Expression
raleqbi1dv	⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B ψ))

Distinct variable groups:   x,A   x,B

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

Proof of Theorem raleqbi1dv

Step	Hyp	Ref	Expression
1		raleq 2808	. 2 ⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B φ))
2		raleqd.1	. . 3 ⊢ (A = B → (φ ↔ ψ))
3	2	ralbidv 2635	. 2 ⊢ (A = B → (∀x ∈ B φ ↔ ∀x ∈ B ψ))
4	1, 3	bitrd 244	1 ⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B ψ))

Syntax hints:   → wi 4   ↔ 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:  peano5  4410  ncfinraise  4482  nnadjoin  4521  nnpweq  4524  tfinnn  4535  spfininduct  4541  isoeq4  5486  trd  5922  extd  5924  symd  5925  trrd  5926  antird  5929  antid  5930  connexrd  5931  connexd  5932  iserd  5943