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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 7164 | . . . 4 ⊢ (𝐵 = 𝑋 → (𝐴 · 𝐵) = (𝐴 · 𝑋)) | |
3 | 2 | eqeq1d 2823 | . . 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 3197 | 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 208 ∧ wa 398 = wceq 1537 ∀wral 3138 (class class class)co 7156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 |
This theorem is referenced by: mgmidmo 17870 ismgmid 17875 ismgmid2 17878 mgmidsssn0 17882 gsumvalx 17886 gsumress 17892 sgrpidmnd 17916 ismndd 17933 mnd1 17952 gsumvallem2 17998 mhmmnd 18221 rngurd 30857 signsw0g 31826 signswmnd 31827 exidu1 35149 cmpidelt 35152 exidres 35171 exidresid 35172 isrngod 35191 rngoideu 35196 zlidlring 44248 2zrngamnd 44261 |
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