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| Mirrors > Home > MPE Home > Th. List > ovanraleqv | Structured version Visualization version GIF version | ||
| Description: Equality theorem for a conjunction with an operation values within a restricted universal quantification. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.) |
| Ref | Expression |
|---|---|
| ovanraleqv.1 | ⊢ (𝐵 = 𝑋 → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| ovanraleqv | ⊢ (𝐵 = 𝑋 → (∀𝑥 ∈ 𝑉 (𝜑 ∧ (𝐴 · 𝐵) = 𝐶) ↔ ∀𝑥 ∈ 𝑉 (𝜓 ∧ (𝐴 · 𝑋) = 𝐶))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ovanraleqv.1 | . . 3 ⊢ (𝐵 = 𝑋 → (𝜑 ↔ 𝜓)) | |
| 2 | oveq2 7439 | . . . 4 ⊢ (𝐵 = 𝑋 → (𝐴 · 𝐵) = (𝐴 · 𝑋)) | |
| 3 | 2 | eqeq1d 2739 | . . 3 ⊢ (𝐵 = 𝑋 → ((𝐴 · 𝐵) = 𝐶 ↔ (𝐴 · 𝑋) = 𝐶)) |
| 4 | 1, 3 | anbi12d 632 | . 2 ⊢ (𝐵 = 𝑋 → ((𝜑 ∧ (𝐴 · 𝐵) = 𝐶) ↔ (𝜓 ∧ (𝐴 · 𝑋) = 𝐶))) |
| 5 | 4 | ralbidv 3178 | 1 ⊢ (𝐵 = 𝑋 → (∀𝑥 ∈ 𝑉 (𝜑 ∧ (𝐴 · 𝐵) = 𝐶) ↔ ∀𝑥 ∈ 𝑉 (𝜓 ∧ (𝐴 · 𝑋) = 𝐶))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∀wral 3061 (class class class)co 7431 |
| 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-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: mgmidmo 18673 ismgmid 18678 ismgmid2 18681 mgmidsssn0 18685 gsumvalx 18689 gsumress 18695 sgrpidmnd 18752 ismndd 18769 mnd1 18792 gsumvallem2 18847 mhmmnd 19082 ringurd 20182 opprring 20347 pzriprnglem7 21498 pzriprnglem13 21504 signsw0g 34571 signswmnd 34572 exidu1 37863 cmpidelt 37866 exidres 37885 exidresid 37886 isrngod 37905 rngoideu 37910 zlidlring 48150 2zrngamnd 48163 |
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