Theorem dropab1

Description: Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011)

	Ref	Expression
Assertion	dropab1	$$⊢\forall xx=y\to \left\{⟨x,z⟩|\phi \right\}=\left\{⟨y,z⟩|\phi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		opeq1	$$⊢x=y\to ⟨x,z⟩=⟨y,z⟩$$
2	1	sps	$$⊢\forall xx=y\to ⟨x,z⟩=⟨y,z⟩$$
3	2	eqeq2d	$$⊢\forall xx=y\to \left(w=⟨x,z⟩\leftrightarrow w=⟨y,z⟩\right)$$
4	3	anbi1d	$$⊢\forall xx=y\to \left(\left(w=⟨x,z⟩\wedge \phi \right)\leftrightarrow \left(w=⟨y,z⟩\wedge \phi \right)\right)$$
5	4	drex2	$$⊢\forall xx=y\to \left(\exists z\left(w=⟨x,z⟩\wedge \phi \right)\leftrightarrow \exists z\left(w=⟨y,z⟩\wedge \phi \right)\right)$$
6	5	drex1	$$⊢\forall xx=y\to \left(\exists x\exists z\left(w=⟨x,z⟩\wedge \phi \right)\leftrightarrow \exists y\exists z\left(w=⟨y,z⟩\wedge \phi \right)\right)$$
7	6	abbidv	$$⊢\forall xx=y\to \left\{w|\exists x\exists z\left(w=⟨x,z⟩\wedge \phi \right)\right\}=\left\{w|\exists y\exists z\left(w=⟨y,z⟩\wedge \phi \right)\right\}$$
8		df-opab	$$⊢\left\{⟨x,z⟩|\phi \right\}=\left\{w|\exists x\exists z\left(w=⟨x,z⟩\wedge \phi \right)\right\}$$
9		df-opab	$$⊢\left\{⟨y,z⟩|\phi \right\}=\left\{w|\exists y\exists z\left(w=⟨y,z⟩\wedge \phi \right)\right\}$$
10	7 8 9	3eqtr4g	$$⊢\forall xx=y\to \left\{⟨x,z⟩|\phi \right\}=\left\{⟨y,z⟩|\phi \right\}$$