Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1294 Structured version   Visualization version   GIF version

Theorem bnj1294 31405
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 3094 . . 3 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
4 sp 2217 . . . 4 (∀𝑥(𝑥𝐴𝜓) → (𝑥𝐴𝜓))
54impcom 397 . . 3 ((𝑥𝐴 ∧ ∀𝑥(𝑥𝐴𝜓)) → 𝜓)
63, 5sylan2b 588 . 2 ((𝑥𝐴 ∧ ∀𝑥𝐴 𝜓) → 𝜓)
71, 2, 6syl2anc 580 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1651  wcel 2157  wral 3089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-12 2213
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-ral 3094
This theorem is referenced by:  bnj1379  31418  bnj1121  31570  bnj1279  31603  bnj1286  31604  bnj1296  31606  bnj1421  31627  bnj1450  31635  bnj1489  31641  bnj1501  31652  bnj1523  31656
  Copyright terms: Public domain W3C validator