Theorem cbv2 1981

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
cbv2.1	⊢ (φ → Ⅎyψ)
cbv2.2	⊢ (φ → Ⅎxχ)
cbv2.3	⊢ (φ → (x = y → (ψ ↔ χ)))

Assertion

Ref	Expression
cbv2	⊢ (∀x∀yφ → (∀xψ ↔ ∀yχ))

Proof of Theorem cbv2

Step	Hyp	Ref	Expression
1		cbv2.1	. . 3 ⊢ (φ → Ⅎyψ)
2	1	nfrd 1763	. 2 ⊢ (φ → (ψ → ∀yψ))
3		cbv2.2	. . 3 ⊢ (φ → Ⅎxχ)
4	3	nfrd 1763	. 2 ⊢ (φ → (χ → ∀xχ))
5		cbv2.3	. 2 ⊢ (φ → (x = y → (ψ ↔ χ)))
6	2, 4, 5	cbv2h 1980	1 ⊢ (∀x∀yφ → (∀xψ ↔ ∀yχ))

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:  cbvald  2008