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Mirrors > Home > ILE Home > Th. List > 2gencl | GIF version |
Description: Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.) |
Ref | Expression |
---|---|
2gencl.1 | ⊢ (𝐶 ∈ 𝑆 ↔ ∃𝑥 ∈ 𝑅 𝐴 = 𝐶) |
2gencl.2 | ⊢ (𝐷 ∈ 𝑆 ↔ ∃𝑦 ∈ 𝑅 𝐵 = 𝐷) |
2gencl.3 | ⊢ (𝐴 = 𝐶 → (𝜑 ↔ 𝜓)) |
2gencl.4 | ⊢ (𝐵 = 𝐷 → (𝜓 ↔ 𝜒)) |
2gencl.5 | ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → 𝜑) |
Ref | Expression |
---|---|
2gencl | ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2gencl.2 | . . . 4 ⊢ (𝐷 ∈ 𝑆 ↔ ∃𝑦 ∈ 𝑅 𝐵 = 𝐷) | |
2 | df-rex 2454 | . . . 4 ⊢ (∃𝑦 ∈ 𝑅 𝐵 = 𝐷 ↔ ∃𝑦(𝑦 ∈ 𝑅 ∧ 𝐵 = 𝐷)) | |
3 | 1, 2 | bitri 183 | . . 3 ⊢ (𝐷 ∈ 𝑆 ↔ ∃𝑦(𝑦 ∈ 𝑅 ∧ 𝐵 = 𝐷)) |
4 | 2gencl.4 | . . . 4 ⊢ (𝐵 = 𝐷 → (𝜓 ↔ 𝜒)) | |
5 | 4 | imbi2d 229 | . . 3 ⊢ (𝐵 = 𝐷 → ((𝐶 ∈ 𝑆 → 𝜓) ↔ (𝐶 ∈ 𝑆 → 𝜒))) |
6 | 2gencl.1 | . . . . . 6 ⊢ (𝐶 ∈ 𝑆 ↔ ∃𝑥 ∈ 𝑅 𝐴 = 𝐶) | |
7 | df-rex 2454 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝑅 𝐴 = 𝐶 ↔ ∃𝑥(𝑥 ∈ 𝑅 ∧ 𝐴 = 𝐶)) | |
8 | 6, 7 | bitri 183 | . . . . 5 ⊢ (𝐶 ∈ 𝑆 ↔ ∃𝑥(𝑥 ∈ 𝑅 ∧ 𝐴 = 𝐶)) |
9 | 2gencl.3 | . . . . . 6 ⊢ (𝐴 = 𝐶 → (𝜑 ↔ 𝜓)) | |
10 | 9 | imbi2d 229 | . . . . 5 ⊢ (𝐴 = 𝐶 → ((𝑦 ∈ 𝑅 → 𝜑) ↔ (𝑦 ∈ 𝑅 → 𝜓))) |
11 | 2gencl.5 | . . . . . 6 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → 𝜑) | |
12 | 11 | ex 114 | . . . . 5 ⊢ (𝑥 ∈ 𝑅 → (𝑦 ∈ 𝑅 → 𝜑)) |
13 | 8, 10, 12 | gencl 2762 | . . . 4 ⊢ (𝐶 ∈ 𝑆 → (𝑦 ∈ 𝑅 → 𝜓)) |
14 | 13 | com12 30 | . . 3 ⊢ (𝑦 ∈ 𝑅 → (𝐶 ∈ 𝑆 → 𝜓)) |
15 | 3, 5, 14 | gencl 2762 | . 2 ⊢ (𝐷 ∈ 𝑆 → (𝐶 ∈ 𝑆 → 𝜒)) |
16 | 15 | impcom 124 | 1 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1348 ∃wex 1485 ∈ wcel 2141 ∃wrex 2449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-gen 1442 ax-ie2 1487 ax-17 1519 |
This theorem depends on definitions: df-bi 116 df-rex 2454 |
This theorem is referenced by: 3gencl 2764 axaddrcl 7814 axmulrcl 7816 axpre-apti 7834 axpre-mulgt0 7836 uzin2 10938 |
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