Theorem cbv3h 1983

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
cbv3h.1	⊢ (φ → ∀yφ)
cbv3h.2	⊢ (ψ → ∀xψ)
cbv3h.3	⊢ (x = y → (φ → ψ))

Assertion

Ref	Expression
cbv3h	⊢ (∀xφ → ∀yψ)

Proof of Theorem cbv3h

Step	Hyp	Ref	Expression
1		cbv3h.1	. . . 4 ⊢ (φ → ∀yφ)
2	1	a1i 10	. . 3 ⊢ (y = y → (φ → ∀yφ))
3		cbv3h.2	. . . 4 ⊢ (ψ → ∀xψ)
4	3	a1i 10	. . 3 ⊢ (y = y → (ψ → ∀xψ))
5		cbv3h.3	. . . 4 ⊢ (x = y → (φ → ψ))
6	5	a1i 10	. . 3 ⊢ (y = y → (x = y → (φ → ψ)))
7	2, 4, 6	cbv1h 1978	. 2 ⊢ (∀x∀y y = y → (∀xφ → ∀yψ))
8		stdpc6 1687	. 2 ⊢ ∀y y = y
9	7, 8	mpg 1548	1 ⊢ (∀xφ → ∀yψ)

Syntax hints:   → wi 4  ∀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:  cleqh  2450