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Mirrors > Home > MPE Home > Th. List > riotaeqdv | Structured version Visualization version GIF version |
Description: Formula-building deduction for iota. (Contributed by NM, 15-Sep-2011.) |
Ref | Expression |
---|---|
riotaeqdv.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
riotaeqdv | ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | riotaeqdv.1 | . . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | eleq2d 2898 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
3 | 2 | anbi1d 631 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))) |
4 | 3 | iotabidv 6333 | . 2 ⊢ (𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = (℩𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))) |
5 | df-riota 7108 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
6 | df-riota 7108 | . 2 ⊢ (℩𝑥 ∈ 𝐵 𝜓) = (℩𝑥(𝑥 ∈ 𝐵 ∧ 𝜓)) | |
7 | 4, 5, 6 | 3eqtr4g 2881 | 1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ℩cio 6306 ℩crio 7107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-rex 3144 df-uni 4832 df-iota 6308 df-riota 7108 |
This theorem is referenced by: riotaeqbidv 7111 grpinvpropd 18168 funtransport 33487 fvtransport 33488 |
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