Theorem reximdv2

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of Margaris p. 90. (Contributed by NM, 17-Sep-2003)

	Ref	Expression
Hypothesis	reximdv2.1	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) → ( 𝑥 ∈ 𝐵 ∧ 𝜒 ) ) )
Assertion	reximdv2	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → ∃ 𝑥 ∈ 𝐵 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		reximdv2.1	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) → ( 𝑥 ∈ 𝐵 ∧ 𝜒 ) ) )
2	1	eximdv	⊢ ( 𝜑 → ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) → ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜒 ) ) )
3		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
4		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐵 𝜒 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜒 ) )
5	2 3 4	3imtr4g	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → ∃ 𝑥 ∈ 𝐵 𝜒 ) )