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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 3437 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 2735 | . 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 1537 ∈ wcel 2106 Vcvv 3478 ℩crio 7387 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-ss 3980 df-uni 4913 df-iota 6516 df-riota 7388 |
This theorem is referenced by: riotaxfrd 7422 lubfval 18408 glbfval 18421 odulub 18465 oduglb 18467 cnlnadjlem5 32100 cdj3lem3 32467 cdj3lem3b 32469 lshpkrlem1 39092 cdleme25cv 40341 cdlemk35 40895 |
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