![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dedths | Structured version Visualization version GIF version |
Description: A version of weak deduction theorem dedth 4579 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 3772 | . . 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 3787 | . . . . 5 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) | |
4 | ifbi 4543 | . . . . 5 ⊢ (([𝑥 / 𝑥]𝜑 ↔ 𝜑) → if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) = if(𝜑, 𝑥, 𝐵)) | |
5 | dfsbcq 3772 | . . . . 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 230 | . . 3 ⊢ [if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) / 𝑥]𝜓 |
8 | 1, 7 | dedth 4579 | . 2 ⊢ ([𝑥 / 𝑥]𝜑 → [𝑥 / 𝑥]𝜓) |
9 | sbcid 3787 | . 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 205 = wceq 1533 [wsbc 3770 ifcif 4521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-12 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-sbc 3771 df-if 4522 |
This theorem is referenced by: renegclALT 38337 |
Copyright terms: Public domain | W3C validator |