Theorem spesbcdi 35413

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 236	. 2 ⊢ (𝜑 → [𝐴 / 𝑥]𝜒)
4	3	spesbcd 3866	1 ⊢ (𝜑 → ∃𝑥𝜒)

Syntax hints:   → wi 4   ↔ wb 208  ∃wex 1780  [wsbc 3772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3496  df-sbc 3773

This theorem is referenced by: (None)