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| Mirrors > Home > MPE Home > Th. List > Mathboxes > renegeulem | Structured version Visualization version GIF version | ||
| Description: Lemma for renegeu 42400 and similar. Remove a change in bound variables from renegeulemv 42398. (Contributed by Steven Nguyen, 28-Jan-2023.) | 
| Ref | Expression | 
|---|---|
| renegeulemv.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) | 
| renegeulemv.1 | ⊢ (𝜑 → ∃𝑦 ∈ ℝ (𝐵 + 𝑦) = 𝐴) | 
| Ref | Expression | 
|---|---|
| renegeulem | ⊢ (𝜑 → ∃!𝑦 ∈ ℝ (𝐵 + 𝑦) = 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | renegeulemv.b | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
| 2 | renegeulemv.1 | . . . 4 ⊢ (𝜑 → ∃𝑦 ∈ ℝ (𝐵 + 𝑦) = 𝐴) | |
| 3 | 1, 2 | renegeulemv 42398 | . . 3 ⊢ (𝜑 → ∃!𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴) | 
| 4 | reurex 3384 | . . 3 ⊢ (∃!𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴 → ∃𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴) | |
| 5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴) | 
| 6 | 1, 5 | renegeulemv 42398 | 1 ⊢ (𝜑 → ∃!𝑦 ∈ ℝ (𝐵 + 𝑦) = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ∃!wreu 3378 (class class class)co 7431 ℝcr 11154 + caddc 11158 | 
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-addrcl 11216 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 | 
| This theorem is referenced by: renegeu 42400 resubeu 42407 | 
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