Theorem iotabidv

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

	Ref	Expression
Hypothesis	iotabidv.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	iotabidv	⊢ ( 𝜑 → ( ℩ 𝑥 𝜓 ) = ( ℩ 𝑥 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		iotabidv.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2	1	alrimiv	⊢ ( 𝜑 → ∀ 𝑥 ( 𝜓 ↔ 𝜒 ) )
3		iotabi	⊢ ( ∀ 𝑥 ( 𝜓 ↔ 𝜒 ) → ( ℩ 𝑥 𝜓 ) = ( ℩ 𝑥 𝜒 ) )
4	2 3	syl	⊢ ( 𝜑 → ( ℩ 𝑥 𝜓 ) = ( ℩ 𝑥 𝜒 ) )