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Mirrors > Home > ILE Home > Th. List > r19.3rmv | GIF version |
Description: Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 6-Aug-2018.) |
Ref | Expression |
---|---|
r19.3rmv | ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1493 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | r19.3rm 3421 | 1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∃wex 1453 ∈ wcel 1465 ∀wral 2393 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-cleq 2110 df-clel 2113 df-ral 2398 |
This theorem is referenced by: iinconstm 3792 exmidsssnc 4096 cnvpom 5051 ssfilem 6737 diffitest 6749 inffiexmid 6768 ctssexmid 6992 exmidonfinlem 7017 caucvgsrlemasr 7566 resqrexlemgt0 10760 |
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