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| Mirrors > Home > MPE Home > Th. List > Mathboxes > riotaeqbii | Structured version Visualization version GIF version | ||
| Description: Equivalent wff's and equal domains yield equal restricted iotas. Inference version. (Contributed by GG, 1-Sep-2025.) |
| Ref | Expression |
|---|---|
| riotaeqbii.1 | ⊢ 𝐴 = 𝐵 |
| riotaeqbii.2 | ⊢ (𝜑 ↔ 𝜓) |
| Ref | Expression |
|---|---|
| riotaeqbii | ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | riotaeqbii.1 | . . . . 5 ⊢ 𝐴 = 𝐵 | |
| 2 | 1 | eleq2i 2832 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) |
| 3 | riotaeqbii.2 | . . . 4 ⊢ (𝜑 ↔ 𝜓) | |
| 4 | 2, 3 | anbi12i 628 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓)) |
| 5 | 4 | iotabii 6544 | . 2 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = (℩𝑥(𝑥 ∈ 𝐵 ∧ 𝜓)) |
| 6 | df-riota 7386 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 7 | df-riota 7386 | . 2 ⊢ (℩𝑥 ∈ 𝐵 𝜓) = (℩𝑥(𝑥 ∈ 𝐵 ∧ 𝜓)) | |
| 8 | 5, 6, 7 | 3eqtr4i 2774 | 1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ℩cio 6510 ℩crio 7385 |
| 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 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-ss 3967 df-uni 4906 df-iota 6512 df-riota 7386 |
| This theorem is referenced by: riotaeqi 36178 |
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