This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ss2rabi

Description: Inference of restricted abstraction subclass from implication. (Contributed by NM, 14-Oct-1999) Avoid axioms. (Revised by SN, 4-Feb-2025)

		Ref	Expression
	Hypothesis	ss2rabi.1	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) )
	Assertion	ss2rabi	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } ⊆ { 𝑥 ∈ 𝐴 ∣ 𝜓 }

Proof

Step	Hyp	Ref	Expression
1		ss2rabi.1	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) )
2	1	adantl	⊢ ( ( ⊤ ∧ 𝑥 ∈ 𝐴 ) → ( 𝜑 → 𝜓 ) )
3	2	ss2rabdv	⊢ ( ⊤ → { 𝑥 ∈ 𝐴 ∣ 𝜑 } ⊆ { 𝑥 ∈ 𝐴 ∣ 𝜓 } )
4	3	mptru	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } ⊆ { 𝑥 ∈ 𝐴 ∣ 𝜓 }