Theorem cbv1 1160

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

Hypotheses

Ref	Expression
cbv1.1	⊢ (φ → (ψ → ∀yψ))
cbv1.2	⊢ (φ → (χ → ∀xχ))
cbv1.3	⊢ (φ → (x = y → (ψ → χ)))

Assertion

Ref	Expression
cbv1	⊢ (∀x∀yφ → (∀xψ → ∀yχ))

Proof of Theorem cbv1

Step	Hyp	Ref	Expression
1		cbv1.1	. . . . 5 ⊢ (φ → (ψ → ∀yψ))
2	1	a4s 982	. . . 4 ⊢ (∀yφ → (ψ → ∀yψ))
3	2	19.20ii 993	. . 3 ⊢ (∀x∀yφ → (∀xψ → ∀x∀yψ))
4		ax-7 960	. . 3 ⊢ (∀x∀yψ → ∀y∀xψ)
5	3, 4	syl6 22	. 2 ⊢ (∀x∀yφ → (∀xψ → ∀y∀xψ))
6		cbv1.3	. . . . . . . 8 ⊢ (φ → (x = y → (ψ → χ)))
7	6	com23 32	. . . . . . 7 ⊢ (φ → (ψ → (x = y → χ)))
8		cbv1.2	. . . . . . 7 ⊢ (φ → (χ → ∀xχ))
9	7, 8	syl6d 56	. . . . . 6 ⊢ (φ → (ψ → (x = y → ∀xχ)))
10	9	19.20ii 993	. . . . 5 ⊢ (∀xφ → (∀xψ → ∀x(x = y → ∀xχ)))
11		ax-9o 1121	. . . . 5 ⊢ (∀x(x = y → ∀xχ) → χ)
12	10, 11	syl6 22	. . . 4 ⊢ (∀xφ → (∀xψ → χ))
13	12	19.20ii 993	. . 3 ⊢ (∀y∀xφ → (∀y∀xψ → ∀yχ))
14	13	a7s 989	. 2 ⊢ (∀x∀yφ → (∀y∀xψ → ∀yχ))
15	5, 14	syld 27	1 ⊢ (∀x∀yφ → (∀xψ → ∀yχ))

Syntax hints:   → wi 3  ∀wal 952   = wceq 954

This theorem is referenced by:  cbv2 1161  cbv3 1162

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-mp 7  ax-7 960  ax-gen 961  ax-4 971  ax-5o 973  ax-9o 1121

Copyright terms: Public domain