Theorem spesbcdi 38654

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 3845	1 ⊢ (𝜑 → ∃𝑥𝜒)

Syntax hints:   → wi 4   ↔ wb 209  ∃wex 1806  [wsbc 3753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-v 3465  df-sbc 3754

This theorem is referenced by: (None)