Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj132 Structured version   Visualization version   GIF version

Theorem bnj132 30046
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj132.1 (𝜑 ↔ ∃𝑥(𝜓𝜒))
Assertion
Ref Expression
bnj132 (𝜑 ↔ (𝜓 → ∃𝑥𝜒))
Distinct variable group:   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜒(𝑥)

Proof of Theorem bnj132
StepHypRef Expression
1 bnj132.1 . 2 (𝜑 ↔ ∃𝑥(𝜓𝜒))
2 19.37v 1897 . 2 (∃𝑥(𝜓𝜒) ↔ (𝜓 → ∃𝑥𝜒))
31, 2bitri 263 1 (𝜑 ↔ (𝜓 → ∃𝑥𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wex 1695
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
This theorem depends on definitions:  df-bi 196  df-ex 1696
This theorem is referenced by:  bnj996  30279
  Copyright terms: Public domain W3C validator