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Theorem bnj1361 31231
 Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1361.1 (𝜑 → ∀𝑥(𝑥𝐴𝑥𝐵))
Assertion
Ref Expression
bnj1361 (𝜑𝐴𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bnj1361
StepHypRef Expression
1 bnj1361.1 . 2 (𝜑 → ∀𝑥(𝑥𝐴𝑥𝐵))
2 dfss2 3738 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
31, 2sylibr 224 1 (𝜑𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1628   ∈ wcel 2144   ⊆ wss 3721 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-in 3728  df-ss 3735 This theorem is referenced by: (None)
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