Theorem cbveuvw 2605

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. See cbveu 2608 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 2047	. . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
3	1	cbvmovw 2601	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
4	2, 3	anbi12i 630	. 2 ⊢ ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓))
5		df-eu 2568	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
6		df-eu 2568	. 2 ⊢ (∃!𝑦𝜓 ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓))
7	4, 5, 6	3bitr4i 306	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∃wex 1787  ∃*wmo 2537  ∃!weu 2567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-mo 2539  df-eu 2568

This theorem is referenced by:  cbvreuvw  3351