Theorem r19.29d2r

Description: Theorem 19.29 of Margaris p. 90 with two restricted quantifiers, deduction version. (Contributed by Thierry Arnoux, 30-Jan-2017)

	Ref	Expression
Hypotheses	r19.29d2r.1	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 )
	r19.29d2r.2	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜒 )
Assertion	r19.29d2r	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜓 ∧ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		r19.29d2r.1	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 )
2		r19.29d2r.2	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜒 )
3		r19.29	⊢ ( ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜓 ∧ ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝜒 ) → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 𝜓 ∧ ∃ 𝑦 ∈ 𝐵 𝜒 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 𝜓 ∧ ∃ 𝑦 ∈ 𝐵 𝜒 ) )
5		r19.29	⊢ ( ( ∀ 𝑦 ∈ 𝐵 𝜓 ∧ ∃ 𝑦 ∈ 𝐵 𝜒 ) → ∃ 𝑦 ∈ 𝐵 ( 𝜓 ∧ 𝜒 ) )
6	5	reximi	⊢ ( ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 𝜓 ∧ ∃ 𝑦 ∈ 𝐵 𝜒 ) → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜓 ∧ 𝜒 ) )
7	4 6	syl	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜓 ∧ 𝜒 ) )