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Theorem abeq2f 2949
 Description: Equality of a class variable and a class abstraction. In this version, the fact that 𝑥 is a non-free variable in 𝐴 is explicitly stated as a hypothesis. (Contributed by Thierry Arnoux, 11-May-2017.) Avoid ax-13 2379. (Revised by Wolf Lammen, 13-May-2023.)
Hypothesis
Ref Expression
abeq2f.0 𝑥𝐴
Assertion
Ref Expression
abeq2f (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))

Proof of Theorem abeq2f
StepHypRef Expression
1 abeq2f.0 . . 3 𝑥𝐴
2 nfab1 2921 . . 3 𝑥{𝑥𝜑}
31, 2cleqf 2947 . 2 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝜑}))
4 abid 2739 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
54bibi2i 341 . . 3 ((𝑥𝐴𝑥 ∈ {𝑥𝜑}) ↔ (𝑥𝐴𝜑))
65albii 1821 . 2 (∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝜑}) ↔ ∀𝑥(𝑥𝐴𝜑))
73, 6bitri 278 1 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  ∀wal 1536   = wceq 1538   ∈ wcel 2111  {cab 2735  Ⅎwnfc 2899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901 This theorem is referenced by:  rabid2f  3300  mptfnf  6466
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