Theorem cbvald 2008

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2016. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypotheses

Ref	Expression
cbvald.1	⊢ Ⅎyφ
cbvald.2	⊢ (φ → Ⅎyψ)
cbvald.3	⊢ (φ → (x = y → (ψ ↔ χ)))

Assertion

Ref	Expression
cbvald	⊢ (φ → (∀xψ ↔ ∀yχ))

Distinct variable groups:   φ,x   χ,x

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

Proof of Theorem cbvald

Step	Hyp	Ref	Expression
1		cbvald.1	. . . 4 ⊢ Ⅎyφ
2	1	nfri 1762	. . 3 ⊢ (φ → ∀yφ)
3	2	alrimiv 1631	. 2 ⊢ (φ → ∀x∀yφ)
4		cbvald.2	. . 3 ⊢ (φ → Ⅎyψ)
5		nfvd 1620	. . 3 ⊢ (φ → Ⅎxχ)
6		cbvald.3	. . 3 ⊢ (φ → (x = y → (ψ ↔ χ)))
7	4, 5, 6	cbv2 1981	. 2 ⊢ (∀x∀yφ → (∀xψ ↔ ∀yχ))
8	3, 7	syl 15	1 ⊢ (φ → (∀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:  cbvexd  2009  cbvaldva  2010