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Theorem dfss2OLD 3936
Description: Obsolete version of dfss2 3935 as of 16-May-2024. (Contributed by NM, 8-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dfss2OLD (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem dfss2OLD
StepHypRef Expression
1 dfss 3933 . . 3 (𝐴𝐵𝐴 = (𝐴𝐵))
2 df-in 3922 . . . 4 (𝐴𝐵) = {𝑥 ∣ (𝑥𝐴𝑥𝐵)}
32eqeq2i 2750 . . 3 (𝐴 = (𝐴𝐵) ↔ 𝐴 = {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
4 eqab 2878 . . 3 (𝐴 = {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴𝑥𝐵)))
51, 3, 43bitri 297 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴𝑥𝐵)))
6 pm4.71 559 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴 ↔ (𝑥𝐴𝑥𝐵)))
76albii 1822 . 2 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ∀𝑥(𝑥𝐴 ↔ (𝑥𝐴𝑥𝐵)))
85, 7bitr4i 278 1 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wcel 2107  {cab 2714  cin 3914  wss 3915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-in 3922  df-ss 3932
This theorem is referenced by: (None)
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