Theorem ovanraleqv

Description: Equality theorem for a conjunction with an operation values within a restricted universal quantification. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022)

	Ref	Expression
Hypothesis	ovanraleqv.1	⊢ ( 𝐵 = 𝑋 → ( 𝜑 ↔ 𝜓 ) )
Assertion	ovanraleqv	⊢ ( 𝐵 = 𝑋 → ( ∀ 𝑥 ∈ 𝑉 ( 𝜑 ∧ ( 𝐴 · 𝐵 ) = 𝐶 ) ↔ ∀ 𝑥 ∈ 𝑉 ( 𝜓 ∧ ( 𝐴 · 𝑋 ) = 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ovanraleqv.1	⊢ ( 𝐵 = 𝑋 → ( 𝜑 ↔ 𝜓 ) )
2		oveq2	⊢ ( 𝐵 = 𝑋 → ( 𝐴 · 𝐵 ) = ( 𝐴 · 𝑋 ) )
3	2	eqeq1d	⊢ ( 𝐵 = 𝑋 → ( ( 𝐴 · 𝐵 ) = 𝐶 ↔ ( 𝐴 · 𝑋 ) = 𝐶 ) )
4	1 3	anbi12d	⊢ ( 𝐵 = 𝑋 → ( ( 𝜑 ∧ ( 𝐴 · 𝐵 ) = 𝐶 ) ↔ ( 𝜓 ∧ ( 𝐴 · 𝑋 ) = 𝐶 ) ) )
5	4	ralbidv	⊢ ( 𝐵 = 𝑋 → ( ∀ 𝑥 ∈ 𝑉 ( 𝜑 ∧ ( 𝐴 · 𝐵 ) = 𝐶 ) ↔ ∀ 𝑥 ∈ 𝑉 ( 𝜓 ∧ ( 𝐴 · 𝑋 ) = 𝐶 ) ) )