Theorem cbv3hv 1850

Description: Lemma for ax10 1944. Similar to cbv3h 1983. Requires distinct variables but avoids ax-12 1925. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 29-Dec-2017.)

Hypotheses

Ref	Expression
cbv3hv.1	⊢ (φ → ∀yφ)
cbv3hv.2	⊢ (ψ → ∀xψ)
cbv3hv.3	⊢ (x = y → (φ → ψ))

Assertion

Ref	Expression
cbv3hv	⊢ (∀xφ → ∀yψ)

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem cbv3hv

Step	Hyp	Ref	Expression
1		cbv3hv.1	. . 3 ⊢ (φ → ∀yφ)
2	1	alimi 1559	. 2 ⊢ (∀xφ → ∀x∀yφ)
3		a9ev 1656	. . . . . . 7 ⊢ ∃x x = y
4		cbv3hv.3	. . . . . . . 8 ⊢ (x = y → (φ → ψ))
5	4	eximi 1576	. . . . . . 7 ⊢ (∃x x = y → ∃x(φ → ψ))
6	3, 5	ax-mp 5	. . . . . 6 ⊢ ∃x(φ → ψ)
7	6	19.35i 1601	. . . . 5 ⊢ (∀xφ → ∃xψ)
8		cbv3hv.2	. . . . . 6 ⊢ (ψ → ∀xψ)
9	8	19.9h 1780	. . . . 5 ⊢ (∃xψ ↔ ψ)
10	7, 9	sylib 188	. . . 4 ⊢ (∀xφ → ψ)
11	10	alimi 1559	. . 3 ⊢ (∀y∀xφ → ∀yψ)
12	11	a7s 1735	. 2 ⊢ (∀x∀yφ → ∀yψ)
13	2, 12	syl 15	1 ⊢ (∀xφ → ∀yψ)

Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1541

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

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

This theorem is referenced by: (None)