Theorem cbvalvOLD 2419

Description: Obsolete version of cbvalv 2417 as of 11-Sep-2023. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-10 2144. (Revised by Wolf Lammen, 17-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
cbvalv.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvalvOLD	⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥

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

Proof of Theorem cbvalvOLD

Step	Hyp	Ref	Expression
1		ax-5 1910	. . . 4 ⊢ (𝜑 → ∀𝑦𝜑)
2	1	hbal 2173	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
3		cbvalv.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	3	spv 2410	. . 3 ⊢ (∀𝑥𝜑 → 𝜓)
5	2, 4	alrimih 1823	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
6		ax-5 1910	. . . 4 ⊢ (𝜓 → ∀𝑥𝜓)
7	6	hbal 2173	. . 3 ⊢ (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓)
8	3	equcoms 2026	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓))
9	8	bicomd 225	. . . 4 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑))
10	9	spv 2410	. . 3 ⊢ (∀𝑦𝜓 → 𝜑)
11	7, 10	alrimih 1823	. 2 ⊢ (∀𝑦𝜓 → ∀𝑥𝜑)
12	5, 11	impbii 211	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-11 2160  ax-12 2176  ax-13 2389

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-nf 1784

This theorem is referenced by: (None)