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Theorem bnj1019 31956
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj1019 (∃𝑝(𝜃𝜒𝜏𝜂) ↔ (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
Distinct variable groups:   𝜒,𝑝   𝜂,𝑝   𝜃,𝑝
Allowed substitution hint:   𝜏(𝑝)

Proof of Theorem bnj1019
StepHypRef Expression
1 19.42v 1947 . 2 (∃𝑝((𝜃𝜒𝜂) ∧ 𝜏) ↔ ((𝜃𝜒𝜂) ∧ ∃𝑝𝜏))
2 bnj258 31883 . . 3 ((𝜃𝜒𝜏𝜂) ↔ ((𝜃𝜒𝜂) ∧ 𝜏))
32exbii 1841 . 2 (∃𝑝(𝜃𝜒𝜏𝜂) ↔ ∃𝑝((𝜃𝜒𝜂) ∧ 𝜏))
4 df-bnj17 31862 . 2 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) ↔ ((𝜃𝜒𝜂) ∧ ∃𝑝𝜏))
51, 3, 43bitr4i 304 1 (∃𝑝(𝜃𝜒𝜏𝜂) ↔ (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1081  wex 1773  w-bnj17 31861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904
This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1083  df-ex 1774  df-bnj17 31862
This theorem is referenced by:  bnj1018  32139  bnj1020  32140  bnj1021  32141
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