Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > vtocl2 | Structured version Visualization version GIF version |
Description: Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Ref | Expression |
---|---|
vtocl2.1 | ⊢ 𝐴 ∈ V |
vtocl2.2 | ⊢ 𝐵 ∈ V |
vtocl2.3 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) |
vtocl2.4 | ⊢ 𝜑 |
Ref | Expression |
---|---|
vtocl2 | ⊢ 𝜓 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtocl2.2 | . 2 ⊢ 𝐵 ∈ V | |
2 | vtocl2.4 | . . . 4 ⊢ 𝜑 | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑦 = 𝐵 → 𝜑) |
4 | vtocl2.1 | . . . 4 ⊢ 𝐴 ∈ V | |
5 | vtocl2.3 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
6 | 5 | pm5.74da 804 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑦 = 𝐵 → 𝜑) ↔ (𝑦 = 𝐵 → 𝜓))) |
7 | 4, 6, 3 | vtocl 3474 | . . 3 ⊢ (𝑦 = 𝐵 → 𝜓) |
8 | 3, 7 | 2thd 268 | . 2 ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓)) |
9 | 1, 8, 2 | vtocl 3474 | 1 ⊢ 𝜓 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-clel 2816 |
This theorem is referenced by: vtocl3 3477 caovord 7419 sornom 9891 wloglei 11364 ipodrsima 18047 mpfind 21067 mclsppslem 33258 monotoddzzfi 40467 |
Copyright terms: Public domain | W3C validator |