Theorem cbv3 847

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

Hypotheses

Ref	Expression
cbv3.1	⊢ (φ → ∀yφ)
cbv3.2	⊢ (ψ → ∀xψ)
cbv3.3	⊢ (x = y → (φ → ψ))

Assertion

Ref	Expression
cbv3	⊢ (∀xφ → ∀yψ)

Proof of Theorem cbv3

Step	Hyp	Ref	Expression
1		cbv3.1	. . . 4 ⊢ (φ → ∀yφ)
2	1	syl3 18	. . 3 ⊢ ((φ → φ) → (φ → ∀yφ))
3		cbv3.2	. . . 4 ⊢ (ψ → ∀xψ)
4	3	a1i 7	. . 3 ⊢ ((φ → φ) → (ψ → ∀xψ))
5		cbv3.3	. . . 4 ⊢ (x = y → (φ → ψ))
6	5	a1i 7	. . 3 ⊢ ((φ → φ) → (x = y → (φ → ψ)))
7	2, 4, 6	cbv1 845	. 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  ∀wal 672   = weq 797

This theorem is referenced by:  mo 1020

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-9 799

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

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