Theorem cbvexdvaw 2039

Description: Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. Version of cbvexdva 2409 with a disjoint variable condition, requiring fewer axioms. (Contributed by David Moews, 1-May-2017.) Avoid ax-13 2371. (Revised by GG, 10-Jan-2024.) Reduce axiom usage. (Revised by Wolf Lammen, 10-Feb-2024.)

Hypothesis

Ref	Expression
cbvaldvaw.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
cbvexdvaw	⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Distinct variable groups:   𝜓,𝑦   𝜒,𝑥   𝜑,𝑥,𝑦

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

Proof of Theorem cbvexdvaw

Step	Hyp	Ref	Expression
1		cbvaldvaw.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
2	1	notbid 318	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (¬ 𝜓 ↔ ¬ 𝜒))
3	2	cbvaldvaw 2038	. . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 ↔ ∀𝑦 ¬ 𝜒))
4		alnex 1781	. . 3 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
5		alnex 1781	. . 3 ⊢ (∀𝑦 ¬ 𝜒 ↔ ¬ ∃𝑦𝜒)
6	3, 4, 5	3bitr3g 313	. 2 ⊢ (𝜑 → (¬ ∃𝑥𝜓 ↔ ¬ ∃𝑦𝜒))
7	6	con4bid 317	1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538  ∃wex 1779

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 1967  ax-7 2008

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780

This theorem is referenced by:  cbvex2vw  2041  isinf  9214  isinfOLD  9215  cbvoprab123vw  36234  cbvoprab13vw  36236  cbveudavw  36246  cbvopab1davw  36259  cbvopab2davw  36260  cbvopabdavw  36261  cbvoprab1davw  36266  cbvoprab2davw  36267  cbvoprab3davw  36268  cbvoprab123davw  36269  cbvoprab12davw  36270  cbvoprab23davw  36271  cbvoprab13davw  36272  bj-gabeqis  36933  grumnud  44282