Theorem raleqbi1dv 2487

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 2479	. 2 ⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B φ))
2		raleqd.1	. . 3 ⊢ (A = B → (φ ↔ ψ))
3	2	ralbidv 2300	. 2 ⊢ (A = B → (∀x ∈ B φ ↔ ∀x ∈ B ψ))
4	1, 3	bitrd 177	1 ⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B ψ))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1226  ∀wral 2280

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000

This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285

This theorem is referenced by:  peano5  4244  isoeq4  5365  bj-indeq  7291  peano5set  7301  bj-nntrans  7312