Theorem cbveuwOLD 2608

Description: Obsolete version of cbveuw 2607 as of 23-May-2024. (Contributed by NM, 25-Nov-1994.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem cbveuwOLD

Step	Hyp	Ref	Expression
1		cbveuwOLD.1	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	sb8euv 2599	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
3		cbveuwOLD.2	. . . 4 ⊢ Ⅎ𝑥𝜓
4		cbveuwOLD.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	3, 4	sbiev 2312	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
6	5	eubii 2585	. 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃!𝑦𝜓)
7	2, 6	bitri 274	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 205  Ⅎwnf 1787  [wsb 2068  ∃!weu 2568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569

This theorem is referenced by: (None)