| Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | ||
| Mirrors > Home > MPE Home > Th. List > vtocl3 | Structured version Visualization version GIF version | ||
| Description: Implicit substitution of classes for setvar variables. (Contributed by NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) Avoid ax-10 2141 and ax-11 2157. (Revised by GG, 20-Aug-2023.) (Proof shortened by Wolf Lammen, 23-Aug-2023.) | 
| Ref | Expression | 
|---|---|
| vtocl3.1 | ⊢ 𝐴 ∈ V | 
| vtocl3.2 | ⊢ 𝐵 ∈ V | 
| vtocl3.3 | ⊢ 𝐶 ∈ V | 
| vtocl3.4 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) | 
| vtocl3.5 | ⊢ 𝜑 | 
| Ref | Expression | 
|---|---|
| vtocl3 | ⊢ 𝜓 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | vtocl3.3 | . 2 ⊢ 𝐶 ∈ V | |
| 2 | vtocl3.5 | . . . 4 ⊢ 𝜑 | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (𝑧 = 𝐶 → 𝜑) | 
| 4 | vtocl3.1 | . . . 4 ⊢ 𝐴 ∈ V | |
| 5 | vtocl3.2 | . . . 4 ⊢ 𝐵 ∈ V | |
| 6 | vtocl3.4 | . . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) | |
| 7 | 6 | 3expa 1119 | . . . . 5 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) | 
| 8 | 7 | pm5.74da 804 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑧 = 𝐶 → 𝜑) ↔ (𝑧 = 𝐶 → 𝜓))) | 
| 9 | 4, 5, 8, 3 | vtocl2 3566 | . . 3 ⊢ (𝑧 = 𝐶 → 𝜓) | 
| 10 | 3, 9 | 2thd 265 | . 2 ⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓)) | 
| 11 | 1, 10, 2 | vtocl 3558 | 1 ⊢ 𝜓 | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-ex 1780 df-clel 2816 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |