Theorem cbvmodavw 36208

Description: Change bound variable in the at-most-one quantifier. Deduction form. (Contributed by GG, 14-Aug-2025.)

Hypothesis

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

Assertion

Ref	Expression
cbvmodavw	⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑦𝜒))

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

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

Proof of Theorem cbvmodavw

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvmodavw.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
2		equequ1 2024	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))
3	2	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))
4	1, 3	imbi12d 344	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝜓 → 𝑥 = 𝑧) ↔ (𝜒 → 𝑦 = 𝑧)))
5	4	cbvaldvaw 2037	. . 3 ⊢ (𝜑 → (∀𝑥(𝜓 → 𝑥 = 𝑧) ↔ ∀𝑦(𝜒 → 𝑦 = 𝑧)))
6	5	exbidv 1920	. 2 ⊢ (𝜑 → (∃𝑧∀𝑥(𝜓 → 𝑥 = 𝑧) ↔ ∃𝑧∀𝑦(𝜒 → 𝑦 = 𝑧)))
7		df-mo 2543	. 2 ⊢ (∃*𝑥𝜓 ↔ ∃𝑧∀𝑥(𝜓 → 𝑥 = 𝑧))
8		df-mo 2543	. 2 ⊢ (∃*𝑦𝜒 ↔ ∃𝑧∀𝑦(𝜒 → 𝑦 = 𝑧))
9	6, 7, 8	3bitr4g 314	1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535  ∃wex 1777  ∃*wmo 2541

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

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-mo 2543

This theorem is referenced by:  cbveudavw  36209  cbvrmodavw  36210  cbvrmodavw2  36241