r19.29ffa

Description: A commonly used pattern based on r19.29 , version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017)

		Ref	Expression
	Hypothesis	r19.29ffa.3	$⊢ (((φ \land x \in A) \land y \in B) \land ψ) \to χ$
	Assertion	r19.29ffa	$⊢ (φ \land \exists x \in A \exists y \in B ψ) \to χ$

Step	Hyp	Ref	Expression
1		r19.29ffa.3	$⊢ (((φ \land x \in A) \land y \in B) \land ψ) \to χ$
2	1	ex	$⊢ ((φ \land x \in A) \land y \in B) \to (ψ \to χ)$
3	2	ralrimiva	$⊢ (φ \land x \in A) \to \forall y \in B (ψ \to χ)$
4	3	ralrimiva	$⊢ φ \to \forall x \in A \forall y \in B (ψ \to χ)$
5	4	adantr	$⊢ (φ \land \exists x \in A \exists y \in B ψ) \to \forall x \in A \forall y \in B (ψ \to χ)$
6		simpr	$⊢ (φ \land \exists x \in A \exists y \in B ψ) \to \exists x \in A \exists y \in B ψ$
7	5 6	r19.29d2r	$⊢ (φ \land \exists x \in A \exists y \in B ψ) \to \exists x \in A \exists y \in B ((ψ \to χ) \land ψ)$
8		pm3.35	$⊢ (ψ \land (ψ \to χ)) \to χ$
9	8	ancoms	$⊢ ((ψ \to χ) \land ψ) \to χ$
10	9	rexlimivw	$⊢ \exists y \in B ((ψ \to χ) \land ψ) \to χ$
11	10	rexlimivw	$⊢ \exists x \in A \exists y \in B ((ψ \to χ) \land ψ) \to χ$
12	7 11	syl	$⊢ (φ \land \exists x \in A \exists y \in B ψ) \to χ$

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