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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 2825 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
3 | 2 | anbi1d 633 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))) |
4 | 3 | iotabidv 6384 | . 2 ⊢ (𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = (℩𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))) |
5 | df-riota 7191 | . 2 ⊢ (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
6 | df-riota 7191 | . 2 ⊢ (℩𝑥 ∈ 𝐵 𝜓) = (℩𝑥(𝑥 ∈ 𝐵 ∧ 𝜓)) | |
7 | 4, 5, 6 | 3eqtr4g 2805 | 1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ℩cio 6356 ℩crio 7190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2717 df-cleq 2731 df-clel 2818 df-v 3425 df-in 3890 df-ss 3900 df-uni 4836 df-iota 6358 df-riota 7191 |
This theorem is referenced by: riotaeqbidv 7194 grpinvpropd 18470 funtransport 34103 fvtransport 34104 |
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