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Theorem spesbcdi 37500
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 233 . 2 (𝜑[𝐴 / 𝑥]𝜒)
43spesbcd 3872 1 (𝜑 → ∃𝑥𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wex 1773  [wsbc 3772
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-v 3470  df-sbc 3773
This theorem is referenced by: (None)
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