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Mirrors > Home > MPE Home > Th. List > Mathboxes > elimhyps2 | Structured version Visualization version GIF version |
Description: Generalization of elimhyps 34247 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 3437 | . 2 ⊢ (𝐴 = if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) → ([𝐴 / 𝑥]𝜑 ↔ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑)) | |
2 | dfsbcq 3437 | . 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 4146 | 1 ⊢ [if([𝐴 / 𝑥]𝜑, 𝐴, 𝐵) / 𝑥]𝜑 |
Colors of variables: wff setvar class |
Syntax hints: [wsbc 3435 ifcif 4086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-sbc 3436 df-if 4087 |
This theorem is referenced by: (None) |
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