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Mirrors > Home > MPE Home > Th. List > vtoclf | Structured version Visualization version GIF version |
Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar 2394. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 26-Jan-2025.) |
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 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | vtoclf.2 | . 2 ⊢ 𝐴 ∈ V | |
3 | vtoclf.4 | . . 3 ⊢ 𝜑 | |
4 | vtoclf.3 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | mpbii 232 | . 2 ⊢ (𝑥 = 𝐴 → 𝜓) |
6 | 1, 2, 5 | vtoclef 3518 | 1 ⊢ 𝜓 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 Vcvv 3448 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-nf 1787 df-clel 2815 |
This theorem is referenced by: vtoclALT 3522 summolem2a 15607 prodmolem2a 15824 poimirlem24 36131 poimirlem28 36135 monotuz 41294 oddcomabszz 41297 binomcxplemnotnn0 42710 limclner 43966 climinf2mpt 44029 climinfmpt 44030 dvnmptdivc 44253 dvnmul 44258 salpreimagtge 45040 salpreimaltle 45041 |
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