Theorem cbv2h 1980

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
cbv2h.1	⊢ (φ → (ψ → ∀yψ))
cbv2h.2	⊢ (φ → (χ → ∀xχ))
cbv2h.3	⊢ (φ → (x = y → (ψ ↔ χ)))

Assertion

Ref	Expression
cbv2h	⊢ (∀x∀yφ → (∀xψ ↔ ∀yχ))

Proof of Theorem cbv2h

Step	Hyp	Ref	Expression
1		cbv2h.1	. . 3 ⊢ (φ → (ψ → ∀yψ))
2		cbv2h.2	. . 3 ⊢ (φ → (χ → ∀xχ))
3		cbv2h.3	. . . 4 ⊢ (φ → (x = y → (ψ ↔ χ)))
4		bi1 178	. . . 4 ⊢ ((ψ ↔ χ) → (ψ → χ))
5	3, 4	syl6 29	. . 3 ⊢ (φ → (x = y → (ψ → χ)))
6	1, 2, 5	cbv1h 1978	. 2 ⊢ (∀x∀yφ → (∀xψ → ∀yχ))
7		equcomi 1679	. . . . 5 ⊢ (y = x → x = y)
8		bi2 189	. . . . 5 ⊢ ((ψ ↔ χ) → (χ → ψ))
9	7, 3, 8	syl56 30	. . . 4 ⊢ (φ → (y = x → (χ → ψ)))
10	2, 1, 9	cbv1h 1978	. . 3 ⊢ (∀y∀xφ → (∀yχ → ∀xψ))
11	10	a7s 1735	. 2 ⊢ (∀x∀yφ → (∀yχ → ∀xψ))
12	6, 11	impbid 183	1 ⊢ (∀x∀yφ → (∀xψ ↔ ∀yχ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  cbv2  1981  eujustALT  2207