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Mirrors > Home > MPE Home > Th. List > Mathboxes > elimhyps2 | Structured version Visualization version GIF version |
Description: Generalization of elimhyps 38344 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 3774 | . 2 ⊢ (𝐴 = if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) → ([𝐴 / 𝑥]𝜑 ↔ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑)) | |
2 | dfsbcq 3774 | . 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 4588 | 1 ⊢ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑 |
Colors of variables: wff setvar class |
Syntax hints: [wsbc 3772 ifcif 4523 |
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-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-sbc 3773 df-if 4524 |
This theorem is referenced by: (None) |
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