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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 7370 | . 2 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒)) |
3 | riotaeqbidv.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
4 | 3 | riotaeqdv 7369 | . 2 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜒) = (℩𝑥 ∈ 𝐵 𝜒)) |
5 | 2, 4 | eqtrd 2771 | 1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1540 ℩crio 7367 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-in 3955 df-ss 3965 df-uni 4909 df-iota 6495 df-riota 7368 |
This theorem is referenced by: dfoi 9510 oieq1 9511 oieq2 9512 ordtypecbv 9516 ordtypelem3 9519 zorn2lem1 10495 zorn2g 10502 cidfval 17625 cidval 17626 cidpropd 17659 lubfval 18308 glbfval 18321 grpinvfval 18900 grpinvfvalALT 18901 pj1fval 19604 mpfrcl 21868 evlsval 21869 q1pval 25907 ig1pval 25926 scutval 27539 mirval 28174 midf 28295 ismidb 28297 lmif 28304 islmib 28306 gidval 30033 grpoinvfval 30043 pjhfval 30917 cvmliftlem5 34579 cvmliftlem15 34588 trlfset 39335 dicffval 40349 dicfval 40350 dihffval 40405 dihfval 40406 hvmapffval 40933 hvmapfval 40934 hdmap1fval 40971 hdmapffval 41001 hdmapfval 41002 hgmapfval 41061 wessf1ornlem 44183 |
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