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Metamath Proof Explorer

Theorem ss2abi

Description: Inference of abstraction subclass from implication. (Contributed by NM, 31-Mar-1995) Avoid ax-8 , ax-10 , ax-11 , ax-12 . (Revised by GG, 28-Jun-2024)

		Ref	Expression
	Hypothesis	ss2abi.1	⊢ ( 𝜑 → 𝜓 )
	Assertion	ss2abi	⊢ { 𝑥 ∣ 𝜑 } ⊆ { 𝑥 ∣ 𝜓 }

Proof

Step	Hyp	Ref	Expression
1		ss2abi.1	⊢ ( 𝜑 → 𝜓 )
2		tru	⊢ ⊤
3	1	a1i	⊢ ( ⊤ → ( 𝜑 → 𝜓 ) )
4	3	ss2abdv	⊢ ( ⊤ → { 𝑥 ∣ 𝜑 } ⊆ { 𝑥 ∣ 𝜓 } )
5	2 4	ax-mp	⊢ { 𝑥 ∣ 𝜑 } ⊆ { 𝑥 ∣ 𝜓 }