Theorem cbveuvw 2606

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. See cbveu 2609 for a version with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 25-Nov-1994.) (Revised by Gino Giotto, 30-Sep-2024.)

Hypothesis

Ref	Expression
cbveuvw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbveuvw	⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

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

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

Proof of Theorem cbveuvw

Step	Hyp	Ref	Expression
1		cbveuvw.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	cbvexvw 2040	. . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
3	1	cbvmovw 2602	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
4	2, 3	anbi12i 627	. 2 ⊢ ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓))
5		df-eu 2569	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
6		df-eu 2569	. 2 ⊢ (∃!𝑦𝜓 ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓))
7	4, 5, 6	3bitr4i 303	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∃wex 1782  ∃*wmo 2538  ∃!weu 2568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-mo 2540  df-eu 2569

This theorem is referenced by:  cbvreuvw  3386