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Theorem abbiOLD 2893
 Description: Obsolete proof of abbi 2826 as of 7-Jan-2024. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
abbiOLD (∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})

Proof of Theorem abbiOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 hbab1 2744 . . 3 (𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
2 hbab1 2744 . . 3 (𝑦 ∈ {𝑥𝜓} → ∀𝑥 𝑦 ∈ {𝑥𝜓})
31, 2cleqh 2876 . 2 ({𝑥𝜑} = {𝑥𝜓} ↔ ∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜓}))
4 abid 2740 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
5 abid 2740 . . . 4 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
64, 5bibi12i 344 . . 3 ((𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜓}) ↔ (𝜑𝜓))
76albii 1822 . 2 (∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜓}) ↔ ∀𝑥(𝜑𝜓))
83, 7bitr2i 279 1 (∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  ∀wal 1537   = wceq 1539   ∈ wcel 2112  {cab 2736 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-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 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-clab 2737  df-cleq 2751  df-clel 2831 This theorem is referenced by: (None)
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