Theorem cbvex1v 35050

Description: Rule used to change bound variables, using implicit substitution. (Contributed by BTernaryTau, 31-Jul-2025.)

Hypotheses

Ref	Expression
cbvex1v.1	⊢ Ⅎ𝑥𝜑
cbvex1v.2	⊢ Ⅎ𝑦𝜑
cbvex1v.3	⊢ (𝜑 → Ⅎ𝑦𝜓)
cbvex1v.4	⊢ (𝜑 → Ⅎ𝑥𝜒)
cbvex1v.5	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))

Assertion

Ref	Expression
cbvex1v	⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑦𝜒))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem cbvex1v

Step	Hyp	Ref	Expression
1		cbvex1v.2	. . . 4 ⊢ Ⅎ𝑦𝜑
2		cbvex1v.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3		cbvex1v.4	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒)
4	3	nfnd 1857	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜒)
5		cbvex1v.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝜓)
6	5	nfnd 1857	. . . 4 ⊢ (𝜑 → Ⅎ𝑦 ¬ 𝜓)
7		equcomi 2016	. . . . 5 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
8		cbvex1v.5	. . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))
9		con3 153	. . . . 5 ⊢ ((𝜓 → 𝜒) → (¬ 𝜒 → ¬ 𝜓))
10	7, 8, 9	syl56 36	. . . 4 ⊢ (𝜑 → (𝑦 = 𝑥 → (¬ 𝜒 → ¬ 𝜓)))
11	1, 2, 4, 6, 10	cbv1v 2342	. . 3 ⊢ (𝜑 → (∀𝑦 ¬ 𝜒 → ∀𝑥 ¬ 𝜓))
12	11	con3d 152	. 2 ⊢ (𝜑 → (¬ ∀𝑥 ¬ 𝜓 → ¬ ∀𝑦 ¬ 𝜒))
13		df-ex 1778	. 2 ⊢ (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓)
14		df-ex 1778	. 2 ⊢ (∃𝑦𝜒 ↔ ¬ ∀𝑦 ¬ 𝜒)
15	12, 13, 14	3imtr4g 296	1 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑦𝜒))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1535  ∃wex 1777  Ⅎwnf 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ex 1778  df-nf 1782

This theorem is referenced by:  dvelimexcased  35053