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Mirrors > Home > MPE Home > Th. List > riotaeqbidv | Structured version Visualization version GIF version |
Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.) |
Ref | Expression |
---|---|
riotaeqbidv.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
riotaeqbidv.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
riotaeqbidv | ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | riotaeqbidv.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | riotabidv 6513 | . 2 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒)) |
3 | riotaeqbidv.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | riotaeqdv 6512 | . 2 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜒) = (℩𝑥 ∈ 𝐵 𝜒)) |
5 | 2, 4 | eqtrd 2644 | 1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ℩crio 6510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-uni 4373 df-iota 5768 df-riota 6511 |
This theorem is referenced by: dfoi 8299 oieq1 8300 oieq2 8301 ordtypecbv 8305 ordtypelem3 8308 zorn2lem1 9201 zorn2g 9208 cidfval 16160 cidval 16161 cidpropd 16193 lubfval 16801 glbfval 16814 grpinvfval 17283 pj1fval 17930 mpfrcl 19339 evlsval 19340 q1pval 23717 ig1pval 23736 mirval 25350 midf 25468 ismidb 25470 lmif 25477 islmib 25479 gidval 26750 grpoinvfval 26760 pjhfval 27639 cvmliftlem5 30525 cvmliftlem15 30534 trlfset 34465 dicffval 35481 dicfval 35482 dihffval 35537 dihfval 35538 hvmapffval 36065 hvmapfval 36066 hdmap1fval 36104 hdmapffval 36136 hdmapfval 36137 hgmapfval 36196 wessf1ornlem 38366 |
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