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Theorem abeq2f 2976
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 2941 . . 3 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
3 hbab1 2795 . . 3 (𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
42, 3cleqh 2908 . 2 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝜑}))
5 abid 2794 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
65bibi2i 328 . . 3 ((𝑥𝐴𝑥 ∈ {𝑥𝜑}) ↔ (𝑥𝐴𝜑))
76albii 1904 . 2 (∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝜑}) ↔ ∀𝑥(𝑥𝐴𝜑))
84, 7bitri 266 1 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wal 1635   = wceq 1637  wcel 2156  {cab 2792  wnfc 2935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937
This theorem is referenced by:  rabid2f  3308  mptfnf  6226
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