Theorem opnneilem 49684

Description: Lemma factoring out common proof steps of opnneil 49688 and opnneirv 49686. (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 3963	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑦))
2	1	adantl 486	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑦))
3		opnneilem.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
4	2, 3	anbi12d 643	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑆 ⊆ 𝑥 ∧ 𝜓) ↔ (𝑆 ⊆ 𝑦 ∧ 𝜒)))
5	4	cbvrexdva 3246	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∃wrex 3089   ⊆ wss 3905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-ss 3922

This theorem is referenced by:  opnneirv  49686  opnneil  49688  opnneieqvv  49690