Theorem cbval 1984

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Hypotheses

Ref	Expression
cbval.1	⊢ Ⅎyφ
cbval.2	⊢ Ⅎxψ
cbval.3	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
cbval	⊢ (∀xφ ↔ ∀yψ)

Proof of Theorem cbval

Step	Hyp	Ref	Expression
1		cbval.1	. . 3 ⊢ Ⅎyφ
2		cbval.2	. . 3 ⊢ Ⅎxψ
3		cbval.3	. . . 4 ⊢ (x = y → (φ ↔ ψ))
4	3	biimpd 198	. . 3 ⊢ (x = y → (φ → ψ))
5	1, 2, 4	cbv3 1982	. 2 ⊢ (∀xφ → ∀yψ)
6	3	biimprd 214	. . . 4 ⊢ (x = y → (ψ → φ))
7	6	equcoms 1681	. . 3 ⊢ (y = x → (ψ → φ))
8	2, 1, 7	cbv3 1982	. 2 ⊢ (∀yψ → ∀xφ)
9	5, 8	impbii 180	1 ⊢ (∀xφ ↔ ∀yψ)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544

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:  cbvex  1985  cbvalv  2002  cbval2  2004  sb8eu  2222  abbi  2464  cleqf  2514  cbvralf  2830  ralab2  3002  cbvralcsf  3199  dfss2f  3265  elintab  3938  cbviota  4345  sb8iota  4347  dffun6f  5124