Theorem cbval 848

Description: Rule used to change bound variables with implicit substitution.

Hypotheses

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

Assertion

Ref	Expression
cbval	⊢ (∀xφ ↔ ∀yψ)

Proof of Theorem cbval

Step	Hyp	Ref	Expression
1		cbval.1	. . . 4 ⊢ (φ → ∀yφ)
2	1	syl3 18	. . 3 ⊢ ((φ → φ) → (φ → ∀yφ))
3		cbval.2	. . . 4 ⊢ (ψ → ∀xψ)
4	3	a1i 7	. . 3 ⊢ ((φ → φ) → (ψ → ∀xψ))
5		cbval.3	. . . 4 ⊢ (x = y → (φ ↔ ψ))
6	5	a1i 7	. . 3 ⊢ ((φ → φ) → (x = y → (φ ↔ ψ)))
7	2, 4, 6	cbv2 846	. 2 ⊢ (∀x∀y(φ → φ) → (∀xφ ↔ ∀yψ))
8		id 9	. . 3 ⊢ (φ → φ)
9	8	ax-gen 677	. 2 ⊢ ∀y(φ → φ)
10	7, 9	mpg 684	1 ⊢ (∀xφ ↔ ∀yψ)

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   = weq 797

This theorem is referenced by:  cbvex 849  cbvalv 972  cbval2 974  cbvald 977  cleqf 1167  cbvralf 1330  dfss2f 1499  elintab 1976  dffunmof 2678  aceq1 3552  nnwof 4609

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-12 802

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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