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Theorem bnj1101 31187
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1101.1 𝑥(𝜑𝜓)
bnj1101.2 (𝜒𝜑)
Assertion
Ref Expression
bnj1101 𝑥(𝜒𝜓)

Proof of Theorem bnj1101
StepHypRef Expression
1 bnj1101.1 . . 3 𝑥(𝜑𝜓)
2 pm3.42 813 . . 3 ((𝜑𝜓) → ((𝜒𝜑) → 𝜓))
31, 2bnj101 31123 . 2 𝑥((𝜒𝜑) → 𝜓)
4 bnj1101.2 . . . . 5 (𝜒𝜑)
54pm4.71i 541 . . . 4 (𝜒 ↔ (𝜒𝜑))
65imbi1i 338 . . 3 ((𝜒𝜓) ↔ ((𝜒𝜑) → 𝜓))
76exbii 1923 . 2 (∃𝑥(𝜒𝜓) ↔ ∃𝑥((𝜒𝜑) → 𝜓))
83, 7mpbir 221 1 𝑥(𝜒𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wex 1851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1852
This theorem is referenced by:  bnj1110  31382  bnj1128  31390  bnj1145  31393
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