Theorem riotabidva

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). ( rabbidva analog.) (Contributed by NM, 17-Jan-2012)

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

Proof

Step	Hyp	Ref	Expression
1		riotabidva.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
2	1	pm5.32da	⊢ ( 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 ) )