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Mirrors > Home > MPE Home > Th. List > Mathboxes > dedths2 | Structured version Visualization version GIF version |
Description: Generalization of dedths 37453 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 3746 | . 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 4549 | 1 ⊢ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsbc 3744 ifcif 4491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-sbc 3745 df-if 4492 |
This theorem is referenced by: (None) |
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