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Mirrors > Home > ILE Home > Th. List > cgsexg | GIF version |
Description: Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.) |
Ref | Expression |
---|---|
cgsexg.1 | ⊢ (𝑥 = 𝐴 → 𝜒) |
cgsexg.2 | ⊢ (𝜒 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cgsexg | ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝜒 ∧ 𝜑) ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cgsexg.2 | . . . 4 ⊢ (𝜒 → (𝜑 ↔ 𝜓)) | |
2 | 1 | biimpa 296 | . . 3 ⊢ ((𝜒 ∧ 𝜑) → 𝜓) |
3 | 2 | exlimiv 1598 | . 2 ⊢ (∃𝑥(𝜒 ∧ 𝜑) → 𝜓) |
4 | elisset 2751 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
5 | cgsexg.1 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝜒) | |
6 | 5 | eximi 1600 | . . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜒) |
7 | 4, 6 | syl 14 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥𝜒) |
8 | 1 | biimprcd 160 | . . . . 5 ⊢ (𝜓 → (𝜒 → 𝜑)) |
9 | 8 | ancld 325 | . . . 4 ⊢ (𝜓 → (𝜒 → (𝜒 ∧ 𝜑))) |
10 | 9 | eximdv 1880 | . . 3 ⊢ (𝜓 → (∃𝑥𝜒 → ∃𝑥(𝜒 ∧ 𝜑))) |
11 | 7, 10 | syl5com 29 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝜓 → ∃𝑥(𝜒 ∧ 𝜑))) |
12 | 3, 11 | impbid2 143 | 1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝜒 ∧ 𝜑) ↔ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∃wex 1492 ∈ wcel 2148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-v 2739 |
This theorem is referenced by: (None) |
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