Theorem cbv3hvOLDOLD 2162

Description: Obsolete proof of cbv3hv 2160 as of 29-Nov-2020. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 29-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
cbv3hv.nf1	⊢ (𝜑 → ∀𝑦𝜑)
cbv3hv.nf2	⊢ (𝜓 → ∀𝑥𝜓)
cbv3hv.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
cbv3hvOLDOLD	⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem cbv3hvOLDOLD

Step	Hyp	Ref	Expression
1		cbv3hv.nf1	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
2	1	alimi 1730	. 2 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑦𝜑)
3		ax6ev 1877	. . . . . . 7 ⊢ ∃𝑥 𝑥 = 𝑦
4		cbv3hv.1	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
5	3, 4	eximii 1754	. . . . . 6 ⊢ ∃𝑥(𝜑 → 𝜓)
6	5	19.35i 1795	. . . . 5 ⊢ (∀𝑥𝜑 → ∃𝑥𝜓)
7		cbv3hv.nf2	. . . . . 6 ⊢ (𝜓 → ∀𝑥𝜓)
8	7	19.9h 2106	. . . . 5 ⊢ (∃𝑥𝜓 ↔ 𝜓)
9	6, 8	sylib 207	. . . 4 ⊢ (∀𝑥𝜑 → 𝜓)
10	9	alimi 1730	. . 3 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑦𝜓)
11	10	alcoms 2022	. 2 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦𝜓)
12	2, 11	syl 17	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701

This theorem is referenced by: (None)