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Mirrors > Home > ILE Home > Th. List > r19.9rmv | GIF version |
Description: Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.) |
Ref | Expression |
---|---|
r19.9rmv | ⊢ (∃y y ∈ A → (φ ↔ ∃x ∈ A φ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2097 | . . 3 ⊢ (𝑎 = y → (𝑎 ∈ A ↔ y ∈ A)) | |
2 | 1 | cbvexv 1792 | . 2 ⊢ (∃𝑎 𝑎 ∈ A ↔ ∃y y ∈ A) |
3 | eleq1 2097 | . . . 4 ⊢ (𝑎 = x → (𝑎 ∈ A ↔ x ∈ A)) | |
4 | 3 | cbvexv 1792 | . . 3 ⊢ (∃𝑎 𝑎 ∈ A ↔ ∃x x ∈ A) |
5 | df-rex 2306 | . . . . 5 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ)) | |
6 | 19.41v 1779 | . . . . 5 ⊢ (∃x(x ∈ A ∧ φ) ↔ (∃x x ∈ A ∧ φ)) | |
7 | 5, 6 | bitri 173 | . . . 4 ⊢ (∃x ∈ A φ ↔ (∃x x ∈ A ∧ φ)) |
8 | 7 | baibr 828 | . . 3 ⊢ (∃x x ∈ A → (φ ↔ ∃x ∈ A φ)) |
9 | 4, 8 | sylbi 114 | . 2 ⊢ (∃𝑎 𝑎 ∈ A → (φ ↔ ∃x ∈ A φ)) |
10 | 2, 9 | sylbir 125 | 1 ⊢ (∃y y ∈ A → (φ ↔ ∃x ∈ A φ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∃wex 1378 ∈ wcel 1390 ∃wrex 2301 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-cleq 2030 df-clel 2033 df-rex 2306 |
This theorem is referenced by: r19.45mv 3309 iunconstm 3656 fconstfvm 5322 ltexprlemloc 6581 |
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