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Mirrors > Home > MPE Home > Th. List > Mathboxes > dedths2 | Structured version Visualization version GIF version |
Description: Generalization of dedths 35978 that is not useful unless we can separately prove ⊢ 𝐴 ∈ V. (Contributed by NM, 13-Jun-2019.) |
Ref | Expression |
---|---|
dedths2.1 | ⊢ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜓 |
Ref | Expression |
---|---|
dedths2 | ⊢ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq 3771 | . 2 ⊢ (𝐴 = if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) → ([𝐴 / 𝑥]𝜓 ↔ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜓)) | |
2 | dedths2.1 | . 2 ⊢ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜓 | |
3 | 1, 2 | dedth 4519 | 1 ⊢ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsbc 3769 ifcif 4463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-sbc 3770 df-if 4464 |
This theorem is referenced by: (None) |
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