Theorem cbveuwOLD 2627

Description: Obsolete version of cbveuw 2626 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 2620	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
3		cbveuwOLD.2	. . . 4 ⊢ Ⅎ𝑥𝜓
4		cbveuwOLD.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	3, 4	sbiev 2323	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
6	5	eubii 2605	. 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃!𝑦𝜓)
7	2, 6	bitri 278	1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 209  Ⅎwnf 1786  [wsb 2070  ∃!weu 2588

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 1912  ax-6 1971  ax-7 2016  ax-10 2143  ax-11 2159  ax-12 2176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589

This theorem is referenced by: (None)