Theorem reximi2

Description: Inference quantifying both antecedent and consequent, based on Theorem 19.22 of Margaris p. 90. (Contributed by NM, 8-Nov-2004)

	Ref	Expression
Hypothesis	reximi2.1	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) )
Assertion	reximi2	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐵 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		reximi2.1	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) )
2	1	eximi	⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) )
3		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
4		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐵 𝜓 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) )
5	2 3 4	3imtr4i	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐵 𝜓 )