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Mirrors > Home > MPE Home > Th. List > Mathboxes > elimhyps | Structured version Visualization version GIF version |
Description: A version of elimhyp 4529 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 3782 | . 2 ⊢ (𝑥 = if(𝜑, 𝑥, 𝐵) → (𝜑 ↔ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑)) | |
2 | dfsbcq 3773 | . 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 4529 | 1 ⊢ [if(𝜑, 𝑥, 𝐵) / 𝑥]𝜑 |
Colors of variables: wff setvar class |
Syntax hints: [wsbc 3771 ifcif 4466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1777 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-sbc 3772 df-if 4467 |
This theorem is referenced by: renegclALT 36093 |
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