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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 3472 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 2821 | . 2 ⊢ V = V | |
2 | riotabiia.1 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | adantl 484 | . . 3 ⊢ ((V = V ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝜓)) |
4 | 3 | riotabidva 7133 | . 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 208 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ℩crio 7113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3943 df-ss 3952 df-uni 4839 df-iota 6314 df-riota 7114 |
This theorem is referenced by: riotaxfrd 7148 lubfval 17588 glbfval 17601 oduglb 17749 odulub 17751 cnlnadjlem5 29848 cdj3lem3 30215 cdj3lem3b 30217 lshpkrlem1 36261 cdleme25cv 37509 cdlemk35 38063 |
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