Theorem spesbcdi 38827

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 3837	1 ⊢ (𝜑 → ∃𝑥𝜒)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∃wex 1812  [wsbc 3746

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-v 3459  df-sbc 3747

This theorem is used by: (None)