Theorem rmoimi2

Description: Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017)

	Ref	Expression
Hypothesis	rmoimi2.1	⊢ ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) )
Assertion	rmoimi2	⊢ ( ∃* 𝑥 ∈ 𝐵 𝜓 → ∃* 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		rmoimi2.1	⊢ ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) )
2		moim	⊢ ( ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) ) → ( ∃* 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) → ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
3	1 2	ax-mp	⊢ ( ∃* 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) → ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
4		df-rmo	⊢ ( ∃* 𝑥 ∈ 𝐵 𝜓 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜓 ) )
5		df-rmo	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜑 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
6	3 4 5	3imtr4i	⊢ ( ∃* 𝑥 ∈ 𝐵 𝜓 → ∃* 𝑥 ∈ 𝐴 𝜑 )