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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 2737 | . . 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 3176 | 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 1537 ∀wral 3059 (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 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 |
This theorem is referenced by: mgmidmo 18686 ismgmid 18691 ismgmid2 18694 mgmidsssn0 18698 gsumvalx 18702 gsumress 18708 sgrpidmnd 18765 ismndd 18782 mnd1 18805 gsumvallem2 18860 mhmmnd 19095 ringurd 20203 opprring 20364 pzriprnglem7 21516 pzriprnglem13 21522 signsw0g 34550 signswmnd 34551 exidu1 37843 cmpidelt 37846 exidres 37865 exidresid 37866 isrngod 37885 rngoideu 37890 zlidlring 48078 2zrngamnd 48091 |
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