Theorem spesbcdi

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

	Ref	Expression
Hypotheses	spesbcdi.1	$$⊢\phi \to \psi$$
	spesbcdi.2	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \leftrightarrow \psi$$
Assertion	spesbcdi	$$⊢\phi \to \exists x\chi$$

Proof

Step	Hyp	Ref	Expression
1		spesbcdi.1	$$⊢\phi \to \psi$$
2		spesbcdi.2	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \leftrightarrow \psi$$
3	1 2	sylibr	$$⊢\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi$$
4	3	spesbcd	$$⊢\phi \to \exists x\chi$$