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Theorem abeq2fOLD 2959
 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.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
abeq2f.0 𝑥𝐴
Assertion
Ref Expression
abeq2fOLD (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))

Proof of Theorem abeq2fOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 abeq2f.0 . . . 4 𝑥𝐴
21nfcrii 2919 . . 3 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
3 hbab1 2760 . . 3 (𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
42, 3cleqh 2883 . 2 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝜑}))
5 abid 2756 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
65bibi2i 330 . . 3 ((𝑥𝐴𝑥 ∈ {𝑥𝜑}) ↔ (𝑥𝐴𝜑))
76albii 1782 . 2 (∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝜑}) ↔ ∀𝑥(𝑥𝐴𝜑))
84, 7bitri 267 1 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198  ∀wal 1505   = wceq 1507   ∈ wcel 2050  {cab 2752  Ⅎwnfc 2910 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912 This theorem is referenced by: (None)
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