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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 𝑦𝜑
Assertion
Ref Expression
eu1OLD (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem eu1OLD
StepHypRef Expression
1 nfs1v 2253 . . 3 𝑥[𝑦 / 𝑥]𝜑
21euf 2594 . 2 (∃!𝑦[𝑦 / 𝑥]𝜑 ↔ ∃𝑥𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥))
3 eu1.nf . . 3 𝑦𝜑
43sb8eu 2634 . 2 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
53sb6rf 2499 . . . . 5 (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))
6 equcom 2064 . . . . . . 7 (𝑥 = 𝑦𝑦 = 𝑥)
76imbi2i 328 . . . . . 6 (([𝑦 / 𝑥]𝜑𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑𝑦 = 𝑥))
87albii 1863 . . . . 5 (∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥))
95, 8anbi12ci 621 . . . 4 ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) ↔ (∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥) ∧ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)))
10 albiim 1935 . . . 4 (∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥) ↔ (∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥) ∧ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)))
119, 10bitr4i 270 . . 3 ((𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥))
1211exbii 1892 . 2 (∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)) ↔ ∃𝑥𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑥))
132, 4, 123bitr4i 295 1 (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑𝑥 = 𝑦)))
 Colors of variables: wff setvar class 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)
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