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Mirrors > Home > ILE Home > Th. List > vtoclf | GIF version |
Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1731. (Contributed by NM, 30-Aug-1993.) |
Ref | Expression |
---|---|
vtoclf.1 | ⊢ Ⅎ𝑥𝜓 |
vtoclf.2 | ⊢ 𝐴 ∈ V |
vtoclf.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
vtoclf.4 | ⊢ 𝜑 |
Ref | Expression |
---|---|
vtoclf | ⊢ 𝜓 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtoclf.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | vtoclf.2 | . . . . 5 ⊢ 𝐴 ∈ V | |
3 | 2 | isseti 2697 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝐴 |
4 | vtoclf.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | biimpd 143 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) |
6 | 3, 5 | eximii 1582 | . . 3 ⊢ ∃𝑥(𝜑 → 𝜓) |
7 | 1, 6 | 19.36i 1651 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
8 | vtoclf.4 | . 2 ⊢ 𝜑 | |
9 | 7, 8 | mpg 1428 | 1 ⊢ 𝜓 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1332 Ⅎwnf 1437 ∈ wcel 1481 Vcvv 2689 |
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-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-v 2691 |
This theorem is referenced by: vtocl 2743 |
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