![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > elimhyps2 | Structured version Visualization version GIF version |
Description: Generalization of elimhyps 37636 that is not useful unless we can separately prove ⊢ 𝐴 ∈ V. (Contributed by NM, 13-Jun-2019.) |
Ref | Expression |
---|---|
elimhyps2.1 | ⊢ [𝐵 / 𝑥]𝜑 |
Ref | Expression |
---|---|
elimhyps2 | ⊢ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq 3775 | . 2 ⊢ (𝐴 = if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) → ([𝐴 / 𝑥]𝜑 ↔ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑)) | |
2 | dfsbcq 3775 | . 2 ⊢ (𝐵 = if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) → ([𝐵 / 𝑥]𝜑 ↔ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑)) | |
3 | elimhyps2.1 | . 2 ⊢ [𝐵 / 𝑥]𝜑 | |
4 | 1, 2, 3 | elimhyp 4587 | 1 ⊢ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑 |
Colors of variables: wff setvar class |
Syntax hints: [wsbc 3773 ifcif 4522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-sbc 3774 df-if 4523 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |