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Metamath Proof Explorer

Theorem r19.29a

Description: A commonly used pattern in the spirit of r19.29 . (Contributed by Thierry Arnoux, 22-Nov-2017) Reduce axiom usage. (Revised by Wolf Lammen, 17-Jun-2023)

	Ref	Expression
Hypotheses	r19.29a.1	$⊢ ((φ \land x \in A) \land ψ) \to χ$
	r19.29a.2	$⊢ φ \to \exists x \in A ψ$
Assertion	r19.29a	$⊢ φ \to χ$

Proof

Step	Hyp	Ref	Expression
1		r19.29a.1	$⊢ ((φ \land x \in A) \land ψ) \to χ$
2		r19.29a.2	$⊢ φ \to \exists x \in A ψ$
3	1	rexlimdva2	$⊢ φ \to (\exists x \in A ψ \to χ)$
4	2 3	mpd	$⊢ φ \to χ$