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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 1640. (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 2563 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝐴 |
4 | vtoclf.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | biimpd 132 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) |
6 | 3, 5 | eximii 1493 | . . 3 ⊢ ∃𝑥(𝜑 → 𝜓) |
7 | 1, 6 | 19.36i 1562 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
8 | vtoclf.4 | . 2 ⊢ 𝜑 | |
9 | 7, 8 | mpg 1340 | 1 ⊢ 𝜓 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 Ⅎwnf 1349 ∈ wcel 1393 Vcvv 2557 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-v 2559 |
This theorem is referenced by: vtocl 2608 |
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