Theorem cbv2 846

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

Hypotheses

Ref	Expression
cbv2.1	⊢ (φ → (ψ → ∀yψ))
cbv2.2	⊢ (φ → (χ → ∀xχ))
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 → (ψ → ∀yψ))
2		cbv2.2	. . 3 ⊢ (φ → (χ → ∀xχ))
3		cbv2.3	. . . 4 ⊢ (φ → (x = y → (ψ ↔ χ)))
4		bi1 130	. . . 4 ⊢ ((ψ ↔ χ) → (ψ → χ))
5	3, 4	syl6 23	. . 3 ⊢ (φ → (x = y → (ψ → χ)))
6	1, 2, 5	cbv1 845	. 2 ⊢ (∀x∀yφ → (∀xψ → ∀yχ))
7		bi2 131	. . . . . 6 ⊢ ((ψ ↔ χ) → (χ → ψ))
8	3, 7	syl6 23	. . . . 5 ⊢ (φ → (x = y → (χ → ψ)))
9		eqcom 811	. . . . 5 ⊢ (y = x → x = y)
10	8, 9	syl5 22	. . . 4 ⊢ (φ → (y = x → (χ → ψ)))
11	2, 1, 10	cbv1 845	. . 3 ⊢ (∀y∀xφ → (∀yχ → ∀xψ))
12	11	a7s 689	. 2 ⊢ (∀x∀yφ → (∀yχ → ∀xψ))
13	6, 12	impbid 397	1 ⊢ (∀x∀yφ → (∀xψ ↔ ∀yχ))

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

This theorem is referenced by:  cbval 848

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 S=r STYLE="color:#FF8C16">802

This theorem depends on definitions:  df-bi 128  df-an198  df-ex 679

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