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Theorem bnj1294 31978
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 3148 . . 3 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
4 sp 2174 . . . 4 (∀𝑥(𝑥𝐴𝜓) → (𝑥𝐴𝜓))
54impcom 408 . . 3 ((𝑥𝐴 ∧ ∀𝑥(𝑥𝐴𝜓)) → 𝜓)
63, 5sylan2b 593 . 2 ((𝑥𝐴 ∧ ∀𝑥𝐴 𝜓) → 𝜓)
71, 2, 6syl2anc 584 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1528  wcel 2107  wral 3143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-12 2169
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-ral 3148
This theorem is referenced by:  bnj1379  31991  bnj1121  32144  bnj1279  32177  bnj1286  32178  bnj1296  32180  bnj1421  32201  bnj1489  32215  bnj1501  32226  bnj1523  32230
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