Theorem cbv2 2403

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2372. See cbv2w 2339 with disjoint variable conditions, not depending on ax-13 2372. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) Format hypotheses to common style, avoid ax-10 2145. (Revised by Wolf Lammen, 10-Sep-2023.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cbv2.1	⊢ Ⅎ𝑥𝜑
cbv2.2	⊢ Ⅎ𝑦𝜑
cbv2.3	⊢ (𝜑 → Ⅎ𝑦𝜓)
cbv2.4	⊢ (𝜑 → Ⅎ𝑥𝜒)
cbv2.5	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
cbv2	⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Proof of Theorem cbv2

Step	Hyp	Ref	Expression
1		cbv2.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		cbv2.2	. . 3 ⊢ Ⅎ𝑦𝜑
3		cbv2.3	. . 3 ⊢ (𝜑 → Ⅎ𝑦𝜓)
4		cbv2.4	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜒)
5		cbv2.5	. . . 4 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
6		biimp 218	. . . 4 ⊢ ((𝜓 ↔ 𝜒) → (𝜓 → 𝜒))
7	5, 6	syl6 35	. . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))
8	1, 2, 3, 4, 7	cbv1 2402	. 2 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
9		equcomi 2029	. . . 4 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
10		biimpr 223	. . . 4 ⊢ ((𝜓 ↔ 𝜒) → (𝜒 → 𝜓))
11	9, 5, 10	syl56 36	. . 3 ⊢ (𝜑 → (𝑦 = 𝑥 → (𝜒 → 𝜓)))
12	2, 1, 4, 3, 11	cbv1 2402	. 2 ⊢ (𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
13	8, 12	impbid 215	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1540  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-11 2162  ax-12 2179  ax-13 2372

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-ex 1787  df-nf 1791

This theorem is referenced by:  cbvald  2407  cbval2  2411  sb9  2523  wl-cbvalnaed  35314  wl-sb8t  35330