Theorem riotabidv

Description: Formula-building deduction for restricted iota. (Contributed by NM, 15-Sep-2011)

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

Proof

Step	Hyp	Ref	Expression
1		riotabidv.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2	1	anbi2d	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜒 ) ) )
3	2	iotabidv	⊢ ( 𝜑 → ( ℩ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) = ( ℩ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜒 ) ) )
4		df-riota	⊢ ( ℩ 𝑥 ∈ 𝐴 𝜓 ) = ( ℩ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
5		df-riota	⊢ ( ℩ 𝑥 ∈ 𝐴 𝜒 ) = ( ℩ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜒 ) )
6	3 4 5	3eqtr4g	⊢ ( 𝜑 → ( ℩ 𝑥 ∈ 𝐴 𝜓 ) = ( ℩ 𝑥 ∈ 𝐴 𝜒 ) )