Theorem eubidOLD 2643

Description: Obsolete proof of eubid 2625 as of 14-Oct-2022. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
eubidOLD.1	⊢ Ⅎ𝑥𝜑
eubidOLD.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
eubidOLD	⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Proof of Theorem eubidOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eubidOLD.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		eubidOLD.2	. . . . 5 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	2	bibi1d 335	. . . 4 ⊢ (𝜑 → ((𝜓 ↔ 𝑥 = 𝑦) ↔ (𝜒 ↔ 𝑥 = 𝑦)))
4	1, 3	albid 2257	. . 3 ⊢ (𝜑 → (∀𝑥(𝜓 ↔ 𝑥 = 𝑦) ↔ ∀𝑥(𝜒 ↔ 𝑥 = 𝑦)))
5	4	exbidv 2017	. 2 ⊢ (𝜑 → (∃𝑦∀𝑥(𝜓 ↔ 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥(𝜒 ↔ 𝑥 = 𝑦)))
6		eu6 2611	. 2 ⊢ (∃!𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 ↔ 𝑥 = 𝑦))
7		eu6 2611	. 2 ⊢ (∃!𝑥𝜒 ↔ ∃𝑦∀𝑥(𝜒 ↔ 𝑥 = 𝑦))
8	5, 6, 7	3bitr4g 306	1 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1651  ∃wex 1875  Ⅎwnf 1879  ∃!weu 2606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-10 2185  ax-11 2200  ax-12 2213

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-ex 1876  df-nf 1880  df-mo 2590  df-eu 2607

This theorem is referenced by: (None)