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Mirrors > Home > ILE Home > Th. List > cgsex2g | GIF version |
Description: Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.) |
Ref | Expression |
---|---|
cgsex2g.1 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜒) |
cgsex2g.2 | ⊢ (𝜒 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cgsex2g | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(𝜒 ∧ 𝜑) ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cgsex2g.2 | . . . 4 ⊢ (𝜒 → (𝜑 ↔ 𝜓)) | |
2 | 1 | biimpa 290 | . . 3 ⊢ ((𝜒 ∧ 𝜑) → 𝜓) |
3 | 2 | exlimivv 1817 | . 2 ⊢ (∃𝑥∃𝑦(𝜒 ∧ 𝜑) → 𝜓) |
4 | elisset 2613 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
5 | elisset 2613 | . . . . . 6 ⊢ (𝐵 ∈ 𝑊 → ∃𝑦 𝑦 = 𝐵) | |
6 | 4, 5 | anim12i 331 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)) |
7 | eeanv 1848 | . . . . 5 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)) | |
8 | 6, 7 | sylibr 132 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) |
9 | cgsex2g.1 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜒) | |
10 | 9 | 2eximi 1532 | . . . 4 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑥∃𝑦𝜒) |
11 | 8, 10 | syl 14 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥∃𝑦𝜒) |
12 | 1 | biimprcd 158 | . . . . 5 ⊢ (𝜓 → (𝜒 → 𝜑)) |
13 | 12 | ancld 318 | . . . 4 ⊢ (𝜓 → (𝜒 → (𝜒 ∧ 𝜑))) |
14 | 13 | 2eximdv 1803 | . . 3 ⊢ (𝜓 → (∃𝑥∃𝑦𝜒 → ∃𝑥∃𝑦(𝜒 ∧ 𝜑))) |
15 | 11, 14 | syl5com 29 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝜓 → ∃𝑥∃𝑦(𝜒 ∧ 𝜑))) |
16 | 3, 15 | impbid2 141 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(𝜒 ∧ 𝜑) ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 ∃wex 1421 ∈ wcel 1433 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-v 2603 |
This theorem is referenced by: (None) |
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