Theorem cbveuALT 2605

Description: Alternative proof of cbveu 2604. Since df-eu 2566 combines two other quantifiers, one can base this theorem on their associated 'change bounded variable' kind of theorems as well. (Contributed by Wolf Lammen, 5-Jan-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cbveu.1	⊢ Ⅎ𝑦𝜑
cbveu.2	⊢ Ⅎ𝑥𝜓
cbveu.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

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

Proof of Theorem cbveuALT

Step	Hyp	Ref	Expression
1		cbveu.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2		cbveu.2	. . . 4 ⊢ Ⅎ𝑥𝜓
3		cbveu.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvex 2401	. . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)
5	1, 2, 3	cbvmo 2601	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
6	4, 5	anbi12i 628	. 2 ⊢ ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓))
7		df-eu 2566	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
8		df-eu 2566	. 2 ⊢ (∃!𝑦𝜓 ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓))
9	6, 7, 8	3bitr4i 303	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1780  Ⅎwnf 1784  ∃*wmo 2535  ∃!weu 2565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-10 2146  ax-11 2162  ax-12 2182  ax-13 2374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566

This theorem is referenced by: (None)