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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 802 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑦 = 𝐵 → 𝜑) ↔ (𝑦 = 𝐵 → 𝜓))) |
7 | 4, 6, 3 | vtocl 3562 | . . 3 ⊢ (𝑦 = 𝐵 → 𝜓) |
8 | 3, 7 | 2thd 267 | . 2 ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓)) |
9 | 1, 8, 2 | vtocl 3562 | 1 ⊢ 𝜓 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-cleq 2817 df-clel 2896 |
This theorem is referenced by: vtocl3 3566 caovord 7362 sornom 9702 wloglei 11175 ipodrsima 17778 mpfind 20323 mclsppslem 32834 monotoddzzfi 39545 |
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