Theorem opnneilem 46151

Description: Lemma factoring out common proof steps of opnneil 46155 and opnneirv 46153. (Contributed by Zhi Wang, 31-Aug-2024.)

Hypothesis

Ref	Expression
opnneilem.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
opnneilem	⊢ (𝜑 → (∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))

Distinct variable groups:   𝑥,𝐽,𝑦   𝑥,𝑆,𝑦   𝜒,𝑥   𝜑,𝑥,𝑦   𝜓,𝑦

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem opnneilem

Step	Hyp	Ref	Expression
1		sseq2 3951	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑦))
2	1	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑦))
3		opnneilem.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
4	2, 3	anbi12d 630	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑆 ⊆ 𝑥 ∧ 𝜓) ↔ (𝑆 ⊆ 𝑦 ∧ 𝜒)))
5	4	cbvrexdva 3392	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∃wrex 3066   ⊆ wss 3891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1544  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-rex 3071  df-v 3432  df-in 3898  df-ss 3908

This theorem is referenced by:  opnneirv  46153  opnneil  46155  opnneieqvv  46157