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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 4587 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 3778 | . . 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 3793 | . . . . 5 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) | |
4 | ifbi 4551 | . . . . 5 ⊢ (([𝑥 / 𝑥]𝜑 ↔ 𝜑) → if([𝑥 / 𝑥]𝜑, 𝑥, 𝐵) = if(𝜑, 𝑥, 𝐵)) | |
5 | dfsbcq 3778 | . . . . 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 4587 | . 2 ⊢ ([𝑥 / 𝑥]𝜑 → [𝑥 / 𝑥]𝜓) |
9 | sbcid 3793 | . 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 1534 [wsbc 3776 ifcif 4529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-12 2167 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-sbc 3777 df-if 4530 |
This theorem is referenced by: renegclALT 38435 |
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