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Theorem abeq2f 2922
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.)
Hypothesis
Ref Expression
abeq2f.0 𝑥𝐴
Assertion
Ref Expression
abeq2f (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))

Proof of Theorem abeq2f
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 abeq2f.0 . . . 4 𝑥𝐴
21nfcrii 2887 . . 3 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
3 hbab1 2741 . . 3 (𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
42, 3cleqh 2854 . 2 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝜑}))
5 abid 2740 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
65bibi2i 326 . . 3 ((𝑥𝐴𝑥 ∈ {𝑥𝜑}) ↔ (𝑥𝐴𝜑))
76albii 1888 . 2 (∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝜑}) ↔ ∀𝑥(𝑥𝐴𝜑))
84, 7bitri 264 1 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1622   = wceq 1624  wcel 2131  {cab 2738  wnfc 2881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883
This theorem is referenced by:  rabid2f  3250  mptfnf  6168
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