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| Mirrors > Home > MPE Home > Th. List > Mathboxes > elimhyps | Structured version Visualization version GIF version | ||
| Description: A version of elimhyp 4591 using explicit substitution. (Contributed by NM, 15-Jun-2019.) | 
| Ref | Expression | 
|---|---|
| elimhyps.1 | ⊢ [𝐵 / 𝑥]𝜑 | 
| Ref | Expression | 
|---|---|
| elimhyps | ⊢ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sbceq1a 3799 | . 2 ⊢ (𝑥 = if(𝜑, 𝑥, 𝐵) → (𝜑 ↔ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑)) | |
| 2 | dfsbcq 3790 | . 2 ⊢ (𝐵 = if(𝜑, 𝑥, 𝐵) → ([𝐵 / 𝑥]𝜑 ↔ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑)) | |
| 3 | elimhyps.1 | . 2 ⊢ [𝐵 / 𝑥]𝜑 | |
| 4 | 1, 2, 3 | elimhyp 4591 | 1 ⊢ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑 | 
| Colors of variables: wff setvar class | 
| Syntax hints: [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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