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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 rule). (rabbiia 3185 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 2622 | . 2 ⊢ V = V | |
2 | riotabiia.1 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | adantl 482 | . . 3 ⊢ ((V = V ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝜓)) |
4 | 3 | riotabidva 6627 | . 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 196 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ℩crio 6610 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-uni 4437 df-iota 5851 df-riota 6611 |
This theorem is referenced by: riotaxfrd 6642 lubfval 16978 glbfval 16991 oduglb 17139 odulub 17141 cnlnadjlem5 28930 cdj3lem3 29297 cdj3lem3b 29299 lshpkrlem1 34397 cdleme25cv 35646 cdlemk35 36200 |
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