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Theorem spesbcdi 38106
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 234 . 2 (𝜑[𝐴 / 𝑥]𝜒)
43spesbcd 3891 1 (𝜑 → ∃𝑥𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wex 1775  [wsbc 3790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-v 3479  df-sbc 3791
This theorem is referenced by: (None)
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