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Theorem abeq1 2280
Description: Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)
Assertion
Ref Expression
abeq1 ({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem abeq1
StepHypRef Expression
1 abeq2 2279 . 2 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
2 eqcom 2172 . 2 ({𝑥𝜑} = 𝐴𝐴 = {𝑥𝜑})
3 bicom 139 . . 3 ((𝜑𝑥𝐴) ↔ (𝑥𝐴𝜑))
43albii 1463 . 2 (∀𝑥(𝜑𝑥𝐴) ↔ ∀𝑥(𝑥𝐴𝜑))
51, 2, 43bitr4i 211 1 ({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
Colors of variables: wff set class
Syntax hints:  wb 104  wal 1346   = wceq 1348  wcel 2141  {cab 2156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166
This theorem is referenced by:  abbi1dv  2290  disj  3462  euabsn2  3650  dm0rn0  4826  dffo3  5640
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