Theorem cbvexdvaw 2041

Description: Rule used to change the bound variable in an existential quantifier with implicit substitution. Deduction form. Version of cbvexdva 2415 with a disjoint variable condition, requiring fewer axioms. (Contributed by David Moews, 1-May-2017.) Avoid ax-13 2377. (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 2040	. . 3 ⊢ (𝜑 → (∀𝑥 ¬ 𝜓 ↔ ∀𝑦 ¬ 𝜒))
4		alnex 1783	. . 3 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
5		alnex 1783	. . 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 1540  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782

This theorem is referenced by:  cbvex2vw  2043  isinf  9169  cbvoprab123vw  36435  cbvoprab13vw  36437  cbveudavw  36447  cbvopab1davw  36460  cbvopab2davw  36461  cbvopabdavw  36462  cbvoprab1davw  36467  cbvoprab2davw  36468  cbvoprab3davw  36469  cbvoprab123davw  36470  cbvoprab12davw  36471  cbvoprab23davw  36472  cbvoprab13davw  36473  bj-gabeqis  37141  grumnud  44594