Theorem ssoprab2i

Description: Inference of operation class abstraction subclass from implication. (Contributed by NM, 11-Nov-1995) (Revised by David Abernethy, 19-Jun-2012)

	Ref	Expression
Hypothesis	ssoprab2i.1	$$⊢\phi \to \psi$$
Assertion	ssoprab2i	$$⊢\left\{⟨⟨x,y⟩,z⟩|\phi \right\}\subseteq \left\{⟨⟨x,y⟩,z⟩|\psi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		ssoprab2i.1	$$⊢\phi \to \psi$$
2	1	anim2i	$$⊢\left(w=⟨x,y⟩\wedge \phi \right)\to \left(w=⟨x,y⟩\wedge \psi \right)$$
3	2	2eximi	$$⊢\exists x\exists y\left(w=⟨x,y⟩\wedge \phi \right)\to \exists x\exists y\left(w=⟨x,y⟩\wedge \psi \right)$$
4	3	ssopab2i	$$⊢\left\{⟨w,z⟩|\exists x\exists y\left(w=⟨x,y⟩\wedge \phi \right)\right\}\subseteq \left\{⟨w,z⟩|\exists x\exists y\left(w=⟨x,y⟩\wedge \psi \right)\right\}$$
5		dfoprab2	$$⊢\left\{⟨⟨x,y⟩,z⟩|\phi \right\}=\left\{⟨w,z⟩|\exists x\exists y\left(w=⟨x,y⟩\wedge \phi \right)\right\}$$
6		dfoprab2	$$⊢\left\{⟨⟨x,y⟩,z⟩|\psi \right\}=\left\{⟨w,z⟩|\exists x\exists y\left(w=⟨x,y⟩\wedge \psi \right)\right\}$$
7	4 5 6	3sstr4i	$$⊢\left\{⟨⟨x,y⟩,z⟩|\phi \right\}\subseteq \left\{⟨⟨x,y⟩,z⟩|\psi \right\}$$