| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 2827 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
| 3 | 2 | anbi1d 631 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))) |
| 4 | 3 | iotabidv 6545 | . 2 ⊢ (𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = (℩𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))) |
| 5 | df-riota 7388 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
| 6 | df-riota 7388 | . 2 ⊢ (℩𝑥 ∈ 𝐵 𝜓) = (℩𝑥(𝑥 ∈ 𝐵 ∧ 𝜓)) | |
| 7 | 4, 5, 6 | 3eqtr4g 2802 | 1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ 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: riotaeqbidv 7391 grpinvpropd 19033 riotaeqbidva 32515 funtransport 36032 fvtransport 36033 |
| Copyright terms: Public domain | W3C validator |