Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1096 Structured version   Visualization version   GIF version

Theorem bnj1096 30107
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1096.1 (𝜑 → ∀𝑥𝜑)
bnj1096.2 (𝜓 ↔ (𝜒𝜃𝜏𝜑))
Assertion
Ref Expression
bnj1096 (𝜓 → ∀𝑥𝜓)
Distinct variable groups:   𝜒,𝑥   𝜏,𝑥   𝜃,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem bnj1096
StepHypRef Expression
1 bnj1096.2 . 2 (𝜓 ↔ (𝜒𝜃𝜏𝜑))
2 ax-5 1827 . . 3 (𝜒 → ∀𝑥𝜒)
3 ax-5 1827 . . 3 (𝜃 → ∀𝑥𝜃)
4 ax-5 1827 . . 3 (𝜏 → ∀𝑥𝜏)
5 bnj1096.1 . . 3 (𝜑 → ∀𝑥𝜑)
62, 3, 4, 5bnj982 30103 . 2 ((𝜒𝜃𝜏𝜑) → ∀𝑥(𝜒𝜃𝜏𝜑))
71, 6hbxfrbi 1742 1 (𝜓 → ∀𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wal 1473  w-bnj17 30005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-bnj17 30006
This theorem is referenced by:  bnj964  30267  bnj981  30274  bnj983  30275  bnj1093  30302  bnj1145  30315
  Copyright terms: Public domain W3C validator