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Theorem spesbcdi 35458
Description: A lemma for introducing an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
Hypotheses
Ref Expression
spesbcdi.1 (𝜑𝜓)
spesbcdi.2 ([𝐴 / 𝑥]𝜒𝜓)
Assertion
Ref Expression
spesbcdi (𝜑 → ∃𝑥𝜒)

Proof of Theorem spesbcdi
StepHypRef Expression
1 spesbcdi.1 . . 3 (𝜑𝜓)
2 spesbcdi.2 . . 3 ([𝐴 / 𝑥]𝜒𝜓)
31, 2sylibr 237 . 2 (𝜑[𝐴 / 𝑥]𝜒)
43spesbcd 3849 1 (𝜑 → ∃𝑥𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wex 1781  [wsbc 3757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-v 3481  df-sbc 3758
This theorem is referenced by: (None)
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