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Mirrors > Home > MPE Home > Th. List > Mathboxes > opnneilem | Structured version Visualization version GIF version |
Description: Lemma factoring out common proof steps of opnneil 48706 and opnneirv 48704. (Contributed by Zhi Wang, 31-Aug-2024.) |
Ref | Expression |
---|---|
opnneilem.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
opnneilem | ⊢ (𝜑 → (∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq2 4022 | . . . 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 632 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑆 ⊆ 𝑥 ∧ 𝜓) ↔ (𝑆 ⊆ 𝑦 ∧ 𝜒))) |
5 | 4 | cbvrexdva 3238 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∃wrex 3068 ⊆ wss 3963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-ss 3980 |
This theorem is referenced by: opnneirv 48704 opnneil 48706 opnneieqvv 48708 |
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