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Theorem bnj133 30778
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj133.1 (𝜑 ↔ ∃𝑥𝜓)
bnj133.2 (𝜒𝜓)
Assertion
Ref Expression
bnj133 (𝜑 ↔ ∃𝑥𝜒)

Proof of Theorem bnj133
StepHypRef Expression
1 bnj133.1 . 2 (𝜑 ↔ ∃𝑥𝜓)
2 bnj133.2 . . 3 (𝜒𝜓)
32exbii 1773 . 2 (∃𝑥𝜒 ↔ ∃𝑥𝜓)
41, 3bitr4i 267 1 (𝜑 ↔ ∃𝑥𝜒)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wex 1703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736
This theorem depends on definitions:  df-bi 197  df-ex 1704
This theorem is referenced by:  bnj150  30931  bnj983  31006  bnj984  31007  bnj985  31008  bnj1090  31032  bnj1514  31116
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