Theorem opnneilem

Description: Lemma factoring out common proof steps of opnneil and opnneirv . (Contributed by Zhi Wang, 31-Aug-2024)

	Ref	Expression
Hypothesis	opnneilem.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	opnneilem	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐽 ( 𝑆 ⊆ 𝑥 ∧ 𝜓 ) ↔ ∃ 𝑦 ∈ 𝐽 ( 𝑆 ⊆ 𝑦 ∧ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		opnneilem.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝜓 ↔ 𝜒 ) )
2		sseq2	⊢ ( 𝑥 = 𝑦 → ( 𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑦 ) )
3	2	adantl	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑦 ) )
4	3 1	anbi12d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( ( 𝑆 ⊆ 𝑥 ∧ 𝜓 ) ↔ ( 𝑆 ⊆ 𝑦 ∧ 𝜒 ) ) )
5	4	cbvrexdva	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐽 ( 𝑆 ⊆ 𝑥 ∧ 𝜓 ) ↔ ∃ 𝑦 ∈ 𝐽 ( 𝑆 ⊆ 𝑦 ∧ 𝜒 ) ) )