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Theorem eqabcb 2883
Description: Equality of a class variable and a class abstraction. Commuted form of eqabb 2881. (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 2881 . 2 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
2 eqcom 2744 . 2 ({𝑥𝜑} = 𝐴𝐴 = {𝑥𝜑})
3 bicom 222 . . 3 ((𝜑𝑥𝐴) ↔ (𝑥𝐴𝜑))
43albii 1819 . 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 1538   = wceq 1540  wcel 2108  {cab 2714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816
This theorem is referenced by:  rabeqcOLD  3690  dm0rn0  5935  dffo3  7122  dffo3f  7126  dfsup2  9484  rankf  9834  fmla0xp  35388  dfon3  35893  dfiota3  35924  onsupmaxb  43251  scottabf  44259
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