Theorem rabbiia

Description: Equivalent formulas yield equal restricted class abstractions (inference form). (Contributed by NM, 22-May-1999)

	Ref	Expression
Hypothesis	rabbiia.1	$$⊢x\in A\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	rabbiia	$$⊢\left\{x\in A|\phi \right\}=\left\{x\in A|\psi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		rabbiia.1	$$⊢x\in A\to \left(\phi \leftrightarrow \psi \right)$$
2	1	pm5.32i	$$⊢\left(x\in A\wedge \phi \right)\leftrightarrow \left(x\in A\wedge \psi \right)$$
3	2	abbii	$$⊢\left\{x|\left(x\in A\wedge \phi \right)\right\}=\left\{x|\left(x\in A\wedge \psi \right)\right\}$$
4		df-rab	$$⊢\left\{x\in A|\phi \right\}=\left\{x|\left(x\in A\wedge \phi \right)\right\}$$
5		df-rab	$$⊢\left\{x\in A|\psi \right\}=\left\{x|\left(x\in A\wedge \psi \right)\right\}$$
6	3 4 5	3eqtr4i	$$⊢\left\{x\in A|\phi \right\}=\left\{x\in A|\psi \right\}$$