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| Mirrors > Home > MPE Home > Th. List > Mathboxes > vtoclefex | Structured version Visualization version GIF version | ||
| Description: Implicit substitution of a class for a setvar variable. (Contributed by ML, 17-Oct-2020.) |
| Ref | Expression |
|---|---|
| vtoclefex.1 | ⊢ Ⅎ𝑥𝜑 |
| vtoclefex.3 | ⊢ (𝑥 = 𝐴 → 𝜑) |
| Ref | Expression |
|---|---|
| vtoclefex | ⊢ (𝐴 ∈ 𝑉 → 𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vtoclefex.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
| 2 | vtoclefex.3 | . . 3 ⊢ (𝑥 = 𝐴 → 𝜑) | |
| 3 | 2 | ax-gen 1795 | . 2 ⊢ ∀𝑥(𝑥 = 𝐴 → 𝜑) |
| 4 | vtoclegft 3587 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴 → 𝜑)) → 𝜑) | |
| 5 | 1, 3, 4 | mp3an23 1455 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 |
| 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 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2714 df-clel 2815 |
| This theorem is referenced by: finxpreclem2 37369 |
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