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Mirrors > Home > MPE Home > Th. List > cgsex2g | Structured version Visualization version 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 500 | . . 3 ⊢ ((𝜒 ∧ 𝜑) → 𝜓) |
3 | 2 | exlimivv 1847 | . 2 ⊢ (∃𝑥∃𝑦(𝜒 ∧ 𝜑) → 𝜓) |
4 | elisset 3188 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
5 | elisset 3188 | . . . . . 6 ⊢ (𝐵 ∈ 𝑊 → ∃𝑦 𝑦 = 𝐵) | |
6 | 4, 5 | anim12i 588 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)) |
7 | eeanv 2170 | . . . . 5 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵)) | |
8 | 6, 7 | sylibr 223 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) |
9 | cgsex2g.1 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜒) | |
10 | 9 | 2eximi 1753 | . . . 4 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑥∃𝑦𝜒) |
11 | 8, 10 | syl 17 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥∃𝑦𝜒) |
12 | 1 | biimprcd 239 | . . . . 5 ⊢ (𝜓 → (𝜒 → 𝜑)) |
13 | 12 | ancld 574 | . . . 4 ⊢ (𝜓 → (𝜒 → (𝜒 ∧ 𝜑))) |
14 | 13 | 2eximdv 1835 | . . 3 ⊢ (𝜓 → (∃𝑥∃𝑦𝜒 → ∃𝑥∃𝑦(𝜒 ∧ 𝜑))) |
15 | 11, 14 | syl5com 31 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝜓 → ∃𝑥∃𝑦(𝜒 ∧ 𝜑))) |
16 | 3, 15 | impbid2 215 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(𝜒 ∧ 𝜑) ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 |
This theorem is referenced by: (None) |
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