Theorem iotabidv

Description: Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011)

	Ref	Expression
Hypothesis	iotabidv.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	iotabidv	$$⊢\phi \to \left(\iota x|\psi \right)=\left(\iota x|\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		iotabidv.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2	1	alrimiv	$$⊢\phi \to \forall x\left(\psi \leftrightarrow \chi \right)$$
3		iotabi	$$⊢\forall x\left(\psi \leftrightarrow \chi \right)\to \left(\iota x|\psi \right)=\left(\iota x|\chi \right)$$
4	2 3	syl	$$⊢\phi \to \left(\iota x|\psi \right)=\left(\iota x|\chi \right)$$