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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dedths | Structured version Visualization version GIF version | ||
| Description: A version of weak deduction theorem dedth 4584 using explicit substitution. (Contributed by NM, 15-Jun-2019.) | 
| Ref | Expression | 
|---|---|
| dedths.1 | ⊢ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜓 | 
| Ref | Expression | 
|---|---|
| dedths | ⊢ (𝜑 → 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfsbcq 3790 | . . 3 ⊢ (𝑥 = if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) → ([𝑥 / 𝑥]𝜓 ↔ [if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) / 𝑥]𝜓)) | |
| 2 | dedths.1 | . . . 4 ⊢ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜓 | |
| 3 | sbcid 3805 | . . . . 5 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) | |
| 4 | ifbi 4548 | . . . . 5 ⊢ (([𝑥 / 𝑥]𝜑 ↔ 𝜑) → if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) = if(𝜑, 𝑥, 𝐵)) | |
| 5 | dfsbcq 3790 | . . . . 5 ⊢ (if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) = if(𝜑, 𝑥, 𝐵) → ([if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) / 𝑥]𝜓 ↔ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜓)) | |
| 6 | 3, 4, 5 | mp2b 10 | . . . 4 ⊢ ([if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) / 𝑥]𝜓 ↔ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜓) | 
| 7 | 2, 6 | mpbir 231 | . . 3 ⊢ [if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) / 𝑥]𝜓 | 
| 8 | 1, 7 | dedth 4584 | . 2 ⊢ ([𝑥 / 𝑥]𝜑 → [𝑥 / 𝑥]𝜓) | 
| 9 | sbcid 3805 | . 2 ⊢ ([𝑥 / 𝑥]𝜓 ↔ 𝜓) | |
| 10 | 8, 3, 9 | 3imtr3i 291 | 1 ⊢ (𝜑 → 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 [wsbc 3788 ifcif 4525 | 
| 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-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-sbc 3789 df-if 4526 | 
| This theorem is referenced by: renegclALT 38964 | 
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