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| Mirrors > Home > MPE Home > Th. List > riotabidva | Structured version Visualization version GIF version | ||
| Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 3443 analog.) (Contributed by NM, 17-Jan-2012.) |
| Ref | Expression |
|---|---|
| riotabidva.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| riotabidva | ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | riotabidva.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | |
| 2 | 1 | pm5.32da 579 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))) |
| 3 | 2 | iotabidv 6545 | . 2 ⊢ (𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))) |
| 4 | df-riota 7388 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
| 5 | df-riota 7388 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜒) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)) | |
| 6 | 3, 4, 5 | 3eqtr4g 2802 | 1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ℩cio 6512 ℩crio 7387 |
| 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-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-uni 4908 df-iota 6514 df-riota 7388 |
| This theorem is referenced by: riotabiia 7408 dfceil2 13879 cidpropd 17753 grpinvpropd 19033 mirval 28663 mirfv 28664 grpoidval 30532 adjval2 31910 riotaeqbidva 32515 xdivval 32901 toslub 32963 tosglb 32965 ringinvval 33239 glbconN 39378 glbconNOLD 39379 cdlemk33N 40911 cdlemk34 40912 cdlemkid4 40936 |
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