Theorem ssrexf

Description: Restricted existential quantification follows from a subclass relationship. (Contributed by Glauco Siliprandi, 20-Apr-2017)

	Ref	Expression
Hypotheses	ssrexf.1	⊢ Ⅎ 𝑥 𝐴
	ssrexf.2	⊢ Ⅎ 𝑥 𝐵
Assertion	ssrexf	⊢ ( 𝐴 ⊆ 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐵 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		ssrexf.1	⊢ Ⅎ 𝑥 𝐴
2		ssrexf.2	⊢ Ⅎ 𝑥 𝐵
3	1 2	nfss	⊢ Ⅎ 𝑥 𝐴 ⊆ 𝐵
4		ssel	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) )
5	4	anim1d	⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) )
6	3 5	eximd	⊢ ( 𝐴 ⊆ 𝐵 → ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ) )
7		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
8		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐵 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) )
9	6 7 8	3imtr4g	⊢ ( 𝐴 ⊆ 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐵 𝜑 ) )