Theorem cbv3 1982

Description: Rule used to change bound variables, using implicit substitution, that does not use ax-12o 2142. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
cbv3.1	⊢ Ⅎyφ
cbv3.2	⊢ Ⅎxψ
cbv3.3	⊢ (x = y → (φ → ψ))

Assertion

Ref	Expression
cbv3	⊢ (∀xφ → ∀yψ)

Proof of Theorem cbv3

Step	Hyp	Ref	Expression
1		cbv3.1	. . . 4 ⊢ Ⅎyφ
2	1	a1i 10	. . 3 ⊢ ( ⊤ → Ⅎyφ)
3		cbv3.2	. . . 4 ⊢ Ⅎxψ
4	3	a1i 10	. . 3 ⊢ ( ⊤ → Ⅎxψ)
5		cbv3.3	. . . 4 ⊢ (x = y → (φ → ψ))
6	5	a1i 10	. . 3 ⊢ ( ⊤ → (x = y → (φ → ψ)))
7	2, 4, 6	cbv1 1979	. 2 ⊢ (∀x∀y ⊤ → (∀xφ → ∀yψ))
8		tru 1321	. . 3 ⊢ ⊤
9	8	ax-gen 1546	. 2 ⊢ ∀y ⊤
10	7, 9	mpg 1548	1 ⊢ (∀xφ → ∀yψ)

Syntax hints:   → wi 4   ⊤ wtru 1316  ∀wal 1540  Ⅎwnf 1544

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:  cbval  1984  ax16i  2046  ax16ALT2  2048  sb8  2092  mo  2226