Theorem cbvex1v 35531

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 1891	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜒)
5		cbvex1v.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝜓)
6	5	nfnd 1891	. . . 4 ⊢ (𝜑 → Ⅎ𝑦 ¬ 𝜓)
7		equcomi 2050	. . . . 5 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
8		cbvex1v.5	. . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))
9		con3 154	. . . . 5 ⊢ ((𝜓 → 𝜒) → (¬ 𝜒 → ¬ 𝜓))
10	7, 8, 9	syl56 37	. . . 4 ⊢ (𝜑 → (𝑦 = 𝑥 → (¬ 𝜒 → ¬ 𝜓)))
11	1, 2, 4, 6, 10	cbv1v 2370	. . 3 ⊢ (𝜑 → (∀𝑦 ¬ 𝜒 → ∀𝑥 ¬ 𝜓))
12	11	con3d 153	. 2 ⊢ (𝜑 → (¬ ∀𝑥 ¬ 𝜓 → ¬ ∀𝑦 ¬ 𝜒))
13		df-ex 1813	. 2 ⊢ (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓)
14		df-ex 1813	. 2 ⊢ (∃𝑦𝜒 ↔ ¬ ∀𝑦 ¬ 𝜒)
15	12, 13, 14	3imtr4g 299	1 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑦𝜒))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568  ∃wex 1812  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817

This theorem is used by:  dvelimexcased  35534