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Mirrors > Home > ILE Home > Th. List > rgenm | GIF version |
Description: Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.) |
Ref | Expression |
---|---|
rgenm.1 | ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝜑) |
Ref | Expression |
---|---|
rgenm | ⊢ ∀𝑥 ∈ 𝐴 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfe1 1385 | . . . . 5 ⊢ Ⅎ𝑥∃𝑥 𝑥 ∈ 𝐴 | |
2 | rgenm.1 | . . . . . 6 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝜑) | |
3 | 2 | ex 108 | . . . . 5 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝜑)) |
4 | 1, 3 | alrimi 1415 | . . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) |
5 | 19.38 1566 | . . . 4 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝜑))) | |
6 | 4, 5 | ax-mp 7 | . . 3 ⊢ ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝜑)) |
7 | pm5.4 238 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝜑)) ↔ (𝑥 ∈ 𝐴 → 𝜑)) | |
8 | 7 | albii 1359 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) |
9 | 6, 8 | mpbi 133 | . 2 ⊢ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑) |
10 | df-ral 2311 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
11 | 9, 10 | mpbir 134 | 1 ⊢ ∀𝑥 ∈ 𝐴 𝜑 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1241 ∃wex 1381 ∈ wcel 1393 ∀wral 2306 |
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 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-ral 2311 |
This theorem is referenced by: (None) |
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