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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 4519 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 3771 | . . 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 3786 | . . . . 5 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) | |
4 | ifbi 4484 | . . . . 5 ⊢ (([𝑥 / 𝑥]𝜑 ↔ 𝜑) → if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) = if(𝜑, 𝑥, 𝐵)) | |
5 | dfsbcq 3771 | . . . . 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 232 | . . 3 ⊢ [if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) / 𝑥]𝜓 |
8 | 1, 7 | dedth 4519 | . 2 ⊢ ([𝑥 / 𝑥]𝜑 → [𝑥 / 𝑥]𝜓) |
9 | sbcid 3786 | . 2 ⊢ ([𝑥 / 𝑥]𝜓 ↔ 𝜓) | |
10 | 8, 3, 9 | 3imtr3i 292 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 [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-12 2167 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: renegclALT 35979 |
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