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Metamath Proof Explorer

Theorem ssopab2i

Description: Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995)

		Ref	Expression
	Hypothesis	ssopab2i.1	$⊢ φ \to ψ$
	Assertion	ssopab2i	$⊢ \{⟨x, y⟩ \| φ\} \subseteq \{⟨x, y⟩ \| ψ\}$

Step	Hyp	Ref	Expression
1		ssopab2i.1	$⊢ φ \to ψ$
2		ssopab2	$⊢ \forall x \forall y (φ \to ψ) \to \{⟨x, y⟩ \| φ\} \subseteq \{⟨x, y⟩ \| ψ\}$
3	1	ax-gen	$⊢ \forall y (φ \to ψ)$
4	2 3	mpg	$⊢ \{⟨x, y⟩ \| φ\} \subseteq \{⟨x, y⟩ \| ψ\}$