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Mirrors > Home > ILE Home > Th. List > gencl | GIF version |
Description: Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.) |
Ref | Expression |
---|---|
gencl.1 | ⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝐵)) |
gencl.2 | ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) |
gencl.3 | ⊢ (𝜒 → 𝜑) |
Ref | Expression |
---|---|
gencl | ⊢ (𝜃 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gencl.1 | . 2 ⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝐵)) | |
2 | gencl.3 | . . . . 5 ⊢ (𝜒 → 𝜑) | |
3 | gencl.2 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | syl5ib 153 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝜒 → 𝜓)) |
5 | 4 | impcom 124 | . . 3 ⊢ ((𝜒 ∧ 𝐴 = 𝐵) → 𝜓) |
6 | 5 | exlimiv 1591 | . 2 ⊢ (∃𝑥(𝜒 ∧ 𝐴 = 𝐵) → 𝜓) |
7 | 1, 6 | sylbi 120 | 1 ⊢ (𝜃 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1348 ∃wex 1485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-gen 1442 ax-ie2 1487 ax-17 1519 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: 2gencl 2763 3gencl 2764 axprecex 7835 axpre-ltirr 7837 |
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