Theorem cbval2 974

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

Hypotheses

Ref	Expression
cbval2.1	⊢ (φ → ∀zφ)
cbval2.2	⊢ (φ → ∀wφ)
cbval2.3	⊢ (ψ → ∀xψ)
cbval2.4	⊢ (ψ → ∀yψ)
cbval2.5	⊢ ((x = z ∧ y = w) → (φ ↔ ψ))

Assertion

Ref	Expression
cbval2	⊢ (∀x∀yφ ↔ ∀z∀wψ)

Distinct variable group(s):   x,y   y,z   x,w   z,w

Proof of Theorem cbval2

Step	Hyp	Ref	Expression
1		cbval2.1	. . 3 ⊢ (φ → ∀zφ)
2	1	hbal 700	. 2 ⊢ (∀yφ → ∀z∀yφ)
3		cbval2.3	. . 3 ⊢ (ψ → ∀xψ)
4	3	hbal 700	. 2 ⊢ (∀wψ → ∀x∀wψ)
5		ax-17 925	. . . . . 6 ⊢ (x = z → ∀w x = z)
6		cbval2.2	. . . . . 6 ⊢ (φ → ∀wφ)
7	5, 6	hban 704	. . . . 5 ⊢ ((x = z ∧ φ) → ∀w(x = z ∧ φ))
8		ax-17 925	. . . . . 6 ⊢ (x = z → ∀y x = z)
9		cbval2.4	. . . . . 6 ⊢ (ψ → ∀yψ)
10	8, 9	hban 704	. . . . 5 ⊢ ((x = z ∧ ψ) → ∀y(x = z ∧ ψ))
11		cbval2.5	. . . . . . . 8 ⊢ ((x = z ∧ y = w) → (φ ↔ ψ))
12	11	exp 291	. . . . . . 7 ⊢ (x = z → (y = w → (φ ↔ ψ)))
13	12	com12 13	. . . . . 6 ⊢ (y = w → (x = z → (φ ↔ ψ)))
14	13	pm5.32d 491	. . . . 5 ⊢ (y = w → ((x = z ∧ φ) ↔ (x = z ∧ ψ)))
15	7, 10, 14	cbval 848	. . . 4 ⊢ (∀y(x = z ∧ φ) ↔ ∀w(x = z ∧ ψ))
16	8	19.28 751	. . . 4 ⊢ (∀y(x = z ∧ φ) ↔ (x = z ∧ ∀yφ))
17	5	19.28 751	. . . 4 ⊢ (∀w(x = z ∧ ψ) ↔ (x = z ∧ ∀wψ))
18	15, 16, 17	3bitr3 156	. . 3 ⊢ ((x = z ∧ ∀yφ) ↔ (x = z ∧ ∀wψ))
19		pm5.32 488	. . 3 ⊢ ((x = z → (∀yφ ↔ ∀wψ)) ↔ ((x = z ∧ ∀yφ) ↔ (x = z ∧ ∀wψ)))
20	18, 19	mpbir 165	. 2 ⊢ (x = z → (∀yφ ↔ ∀wψ))
21	2, 4, 20	cbval 848	1 ⊢ (∀x∀yφ ↔ ∀z∀wψ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   = weq 797

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  ax-17 925

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

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