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Theorem bnj1361 32104
 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 3929 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
31, 2sylibr 236 1 (𝜑𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1535   ∈ wcel 2114   ⊆ wss 3909 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2792 This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-v 3472  df-in 3916  df-ss 3926 This theorem is referenced by: (None)
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