Theorem cbveudavw 36209

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

Hypothesis

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

Assertion

Ref	Expression
cbveudavw	⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑦𝜒))

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

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

Proof of Theorem cbveudavw

Step	Hyp	Ref	Expression
1		cbveudavw.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
2	1	cbvexdvaw 2038	. . 3 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
3	1	cbvmodavw 36208	. . 3 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑦𝜒))
4	2, 3	anbi12d 631	. 2 ⊢ (𝜑 → ((∃𝑥𝜓 ∧ ∃*𝑥𝜓) ↔ (∃𝑦𝜒 ∧ ∃*𝑦𝜒)))
5		df-eu 2572	. 2 ⊢ (∃!𝑥𝜓 ↔ (∃𝑥𝜓 ∧ ∃*𝑥𝜓))
6		df-eu 2572	. 2 ⊢ (∃!𝑦𝜒 ↔ (∃𝑦𝜒 ∧ ∃*𝑦𝜒))
7	4, 5, 6	3bitr4g 314	1 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1777  ∃*wmo 2541  ∃!weu 2571

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  df-eu 2572

This theorem is referenced by:  cbvreudavw  36211  cbvreudavw2  36242