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Theorem bnj1405 31235
 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 4676 . 2 (𝑋 𝑦𝐴 𝐵 ↔ ∃𝑦𝐴 𝑋𝐵)
31, 2sylib 208 1 (𝜑 → ∃𝑦𝐴 𝑋𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2139  ∃wrex 3051  ∪ ciun 4672 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-iun 4674 This theorem is referenced by:  bnj1408  31432  bnj1450  31446  bnj1501  31463
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