Theorem eu1OLD 2642

Description: Obsolete version of eu1 2641 as of 7-Feb-2023. (Contributed by NM, 20-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 29-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
eu1.nf	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
eu1OLD	⊢ (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem eu1OLD

Step	Hyp	Ref	Expression
1		nfs1v 2253	. . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
2	1	euf 2594	. 2 ⊢ (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑥∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑥))
3		eu1.nf	. . 3 ⊢ Ⅎ𝑦𝜑
4	3	sb8eu 2634	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
5	3	sb6rf 2499	. . . . 5 ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
6		equcom 2064	. . . . . . 7 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
7	6	imbi2i 328	. . . . . 6 ⊢ (([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑥))
8	7	albii 1863	. . . . 5 ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑥))
9	5, 8	anbi12ci 621	. . . 4 ⊢ ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)) ↔ (∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑥) ∧ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)))
10		albiim 1935	. . . 4 ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑥) ↔ (∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 = 𝑥) ∧ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)))
11	9, 10	bitr4i 270	. . 3 ⊢ ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑥))
12	11	exbii 1892	. 2 ⊢ (∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)) ↔ ∃𝑥∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑥))
13	2, 4, 12	3bitr4i 295	1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1599  ∃wex 1823  Ⅎwnf 1827  [wsb 2011  ∃!weu 2585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586

This theorem is referenced by: (None)