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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 48807 and opnneirv 48805. (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 4010 | . . . 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 3240 | 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 3070 ⊆ wss 3951 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-ss 3968 | 
| This theorem is referenced by: opnneirv 48805 opnneil 48807 opnneieqvv 48809 | 
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