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| Mirrors > Home > MPE Home > Th. List > riotabiia | Structured version Visualization version GIF version | ||
| Description: Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 3440 analog.) (Contributed by NM, 16-Jan-2012.) |
| Ref | Expression |
|---|---|
| riotabiia.1 | ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| riotabiia | ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐴 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . 2 ⊢ V = V | |
| 2 | riotabiia.1 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) | |
| 3 | 2 | adantl 481 | . . 3 ⊢ ((V = V ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝜓)) |
| 4 | 3 | riotabidva 7407 | . 2 ⊢ (V = V → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐴 𝜓)) |
| 5 | 1, 4 | ax-mp 5 | 1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐴 𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ℩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: riotaxfrd 7422 lubfval 18395 glbfval 18408 odulub 18452 oduglb 18454 cnlnadjlem5 32090 cdj3lem3 32457 cdj3lem3b 32459 lshpkrlem1 39111 cdleme25cv 40360 cdlemk35 40914 |
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