Theorem spesbcdi 38073

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 234	. 2 ⊢ (𝜑 → [𝐴 / 𝑥]𝜒)
4	3	spesbcd 3905	1 ⊢ (𝜑 → ∃𝑥𝜒)

Syntax hints:   → wi 4   ↔ wb 206  ∃wex 1777  [wsbc 3804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-v 3490  df-sbc 3805

This theorem is referenced by: (None)