Theorem cbvalv 2002

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
cbvalv.1	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
cbvalv	⊢ (∀xφ ↔ ∀yψ)

Distinct variable groups:   φ,y   ψ,x

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

Proof of Theorem cbvalv

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎyφ
2		nfv 1619	. 2 ⊢ Ⅎxψ
3		cbvalv.1	. 2 ⊢ (x = y → (φ ↔ ψ))
4	1, 2, 3	cbval 1984	1 ⊢ (∀xφ ↔ ∀yψ)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540

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

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  ax10-16  2190  nfcjust  2478  sfintfin  4533