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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 3425 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 582 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))) |
3 | 2 | iotabidv 6308 | . 2 ⊢ (𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))) |
4 | df-riota 7093 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
5 | df-riota 7093 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜒) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)) | |
6 | 3, 4, 5 | 3eqtr4g 2858 | 1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ℩cio 6281 ℩crio 7092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-ss 3898 df-uni 4801 df-iota 6283 df-riota 7093 |
This theorem is referenced by: riotabiia 7113 dfceil2 13204 cidpropd 16972 grpinvpropd 18166 mirval 26449 mirfv 26450 grpoidval 28296 adjval2 29674 xdivval 30621 toslub 30681 tosglb 30683 ringinvval 30914 glbconN 36673 cdlemk33N 38205 cdlemk34 38206 cdlemkid4 38230 |
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