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Theorem eqabcb 2881
Description: Equality of a class variable and a class abstraction. Commuted form of eqabb 2879. (Contributed by NM, 20-Aug-1993.)
Assertion
Ref Expression
eqabcb ({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem eqabcb
StepHypRef Expression
1 eqabb 2879 . 2 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
2 eqcom 2742 . 2 ({𝑥𝜑} = 𝐴𝐴 = {𝑥𝜑})
3 bicom 222 . . 3 ((𝜑𝑥𝐴) ↔ (𝑥𝐴𝜑))
43albii 1816 . 2 (∀𝑥(𝜑𝑥𝐴) ↔ ∀𝑥(𝑥𝐴𝜑))
51, 2, 43bitr4i 303 1 ({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wal 1535   = wceq 1537  wcel 2106  {cab 2712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814
This theorem is referenced by:  rabeqcOLD  3693  dm0rn0  5938  dffo3  7122  dffo3f  7126  dfsup2  9482  rankf  9832  fmla0xp  35368  dfon3  35874  dfiota3  35905  onsupmaxb  43228  scottabf  44236
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