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Theorem bnj1294 30862
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1294.1 (𝜑 → ∀𝑥𝐴 𝜓)
bnj1294.2 (𝜑𝑥𝐴)
Assertion
Ref Expression
bnj1294 (𝜑𝜓)

Proof of Theorem bnj1294
StepHypRef Expression
1 bnj1294.2 . 2 (𝜑𝑥𝐴)
2 bnj1294.1 . 2 (𝜑 → ∀𝑥𝐴 𝜓)
3 df-ral 2914 . . 3 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
4 sp 2051 . . . 4 (∀𝑥(𝑥𝐴𝜓) → (𝑥𝐴𝜓))
54impcom 446 . . 3 ((𝑥𝐴 ∧ ∀𝑥(𝑥𝐴𝜓)) → 𝜓)
63, 5sylan2b 492 . 2 ((𝑥𝐴 ∧ ∀𝑥𝐴 𝜓) → 𝜓)
71, 2, 6syl2anc 692 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1479  wcel 1988  wral 2909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-12 2045
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703  df-ral 2914
This theorem is referenced by:  bnj1379  30875  bnj1121  31027  bnj1279  31060  bnj1286  31061  bnj1296  31063  bnj1421  31084  bnj1450  31092  bnj1489  31098  bnj1501  31109  bnj1523  31113
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