Theorem raleqbi1dv 2815

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 2807	. 2 ⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B φ))
2		raleqd.1	. . 3 ⊢ (A = B → (φ ↔ ψ))
3	2	ralbidv 2634	. 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 2614

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 2478  df-ral 2619

This theorem is referenced by:  peano5  4409  ncfinraise  4481  nnadjoin  4520  nnpweq  4523  tfinnn  4534  spfininduct  4540  isoeq4  5485  trd  5921  extd  5923  symd  5924  trrd  5925  antird  5928  antid  5929  connexrd  5930  connexd  5931  iserd  5942