Theorem spesbcdi 35558

Description: A lemma for introducing an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)

Hypotheses

Ref	Expression
spesbcdi.1	⊢ (𝜑 → 𝜓)
spesbcdi.2	⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜓)

Assertion

Ref	Expression
spesbcdi	⊢ (𝜑 → ∃𝑥𝜒)

Proof of Theorem spesbcdi

Step	Hyp	Ref	Expression
1		spesbcdi.1	. . 3 ⊢ (𝜑 → 𝜓)
2		spesbcdi.2	. . 3 ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜓)
3	1, 2	sylibr 237	. 2 ⊢ (𝜑 → [𝐴 / 𝑥]𝜒)
4	3	spesbcd 3812	1 ⊢ (𝜑 → ∃𝑥𝜒)

Syntax hints:   → wi 4   ↔ wb 209  ∃wex 1781  [wsbc 3720

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721

This theorem is referenced by: (None)