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Theorem bnj1405 32101
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1405.1 (𝜑𝑋 𝑦𝐴 𝐵)
Assertion
Ref Expression
bnj1405 (𝜑 → ∃𝑦𝐴 𝑋𝐵)
Distinct variable group:   𝑦,𝑋
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑦)

Proof of Theorem bnj1405
StepHypRef Expression
1 bnj1405.1 . 2 (𝜑𝑋 𝑦𝐴 𝐵)
2 eliun 4914 . 2 (𝑋 𝑦𝐴 𝐵 ↔ ∃𝑦𝐴 𝑋𝐵)
31, 2sylib 220 1 (𝜑 → ∃𝑦𝐴 𝑋𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wrex 3137   ciun 4910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-v 3495  df-iun 4912
This theorem is referenced by:  bnj1408  32301  bnj1450  32315  bnj1501  32332
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