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Metamath Proof Explorer

Theorem r19.29d2r

Description: Theorem 19.29 of Margaris p. 90 with two restricted quantifiers, deduction version. (Contributed by Thierry Arnoux, 30-Jan-2017) (Proof shortened by Wolf Lammen, 4-Nov-2024)

	Ref	Expression
Hypotheses	r19.29d2r.1	\|- ( ph -> A. x e. A A. y e. B ps )
	r19.29d2r.2	\|- ( ph -> E. x e. A E. y e. B ch )
Assertion	r19.29d2r	\|- ( ph -> E. x e. A E. y e. B ( ps /\ ch ) )

Proof

Step	Hyp	Ref	Expression
1		r19.29d2r.1	\|- ( ph -> A. x e. A A. y e. B ps )
2		r19.29d2r.2	\|- ( ph -> E. x e. A E. y e. B ch )
3	1 2	jca	\|- ( ph -> ( A. x e. A A. y e. B ps /\ E. x e. A E. y e. B ch ) )
4		2r19.29	\|- ( ( A. x e. A A. y e. B ps /\ E. x e. A E. y e. B ch ) -> E. x e. A E. y e. B ( ps /\ ch ) )
5	3 4	syl	\|- ( ph -> E. x e. A E. y e. B ( ps /\ ch ) )