Theorem oprabbid

Description: Equivalent wff's yield equal operation class abstractions (deduction form). (Contributed by NM, 21-Feb-2004) (Revised by Mario Carneiro, 24-Jun-2014)

	Ref	Expression
Hypotheses	oprabbid.1	$$⊢Ⅎx\phi$$
	oprabbid.2	$$⊢Ⅎy\phi$$
	oprabbid.3	$$⊢Ⅎz\phi$$
	oprabbid.4	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	oprabbid	$$⊢\phi \to \left\{⟨⟨x,y⟩,z⟩|\psi \right\}=\left\{⟨⟨x,y⟩,z⟩|\chi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		oprabbid.1	$$⊢Ⅎx\phi$$
2		oprabbid.2	$$⊢Ⅎy\phi$$
3		oprabbid.3	$$⊢Ⅎz\phi$$
4		oprabbid.4	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
5	4	anbi2d	$$⊢\phi \to \left(\left(w=⟨⟨x,y⟩,z⟩\wedge \psi \right)\leftrightarrow \left(w=⟨⟨x,y⟩,z⟩\wedge \chi \right)\right)$$
6	3 5	exbid	$$⊢\phi \to \left(\exists z\left(w=⟨⟨x,y⟩,z⟩\wedge \psi \right)\leftrightarrow \exists z\left(w=⟨⟨x,y⟩,z⟩\wedge \chi \right)\right)$$
7	2 6	exbid	$$⊢\phi \to \left(\exists y\exists z\left(w=⟨⟨x,y⟩,z⟩\wedge \psi \right)\leftrightarrow \exists y\exists z\left(w=⟨⟨x,y⟩,z⟩\wedge \chi \right)\right)$$
8	1 7	exbid	$$⊢\phi \to \left(\exists x\exists y\exists z\left(w=⟨⟨x,y⟩,z⟩\wedge \psi \right)\leftrightarrow \exists x\exists y\exists z\left(w=⟨⟨x,y⟩,z⟩\wedge \chi \right)\right)$$
9	8	abbidv	$$⊢\phi \to \left\{w|\exists x\exists y\exists z\left(w=⟨⟨x,y⟩,z⟩\wedge \psi \right)\right\}=\left\{w|\exists x\exists y\exists z\left(w=⟨⟨x,y⟩,z⟩\wedge \chi \right)\right\}$$
10		df-oprab	$$⊢\left\{⟨⟨x,y⟩,z⟩|\psi \right\}=\left\{w|\exists x\exists y\exists z\left(w=⟨⟨x,y⟩,z⟩\wedge \psi \right)\right\}$$
11		df-oprab	$$⊢\left\{⟨⟨x,y⟩,z⟩|\chi \right\}=\left\{w|\exists x\exists y\exists z\left(w=⟨⟨x,y⟩,z⟩\wedge \chi \right)\right\}$$
12	9 10 11	3eqtr4g	$$⊢\phi \to \left\{⟨⟨x,y⟩,z⟩|\psi \right\}=\left\{⟨⟨x,y⟩,z⟩|\chi \right\}$$