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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 2412. (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 3511 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝐴 |
4 | vtoclf.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | biimpd 231 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) |
6 | 3, 5 | eximii 1836 | . . 3 ⊢ ∃𝑥(𝜑 → 𝜓) |
7 | 1, 6 | 19.36i 2232 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
8 | vtoclf.4 | . 2 ⊢ 𝜑 | |
9 | 7, 8 | mpg 1797 | 1 ⊢ 𝜓 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 Ⅎwnf 1783 ∈ 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-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-nf 1784 df-cleq 2817 df-clel 2896 |
This theorem is referenced by: vtoclALT 3563 summolem2a 15075 prodmolem2a 15291 poimirlem24 34920 poimirlem28 34924 monotuz 39544 oddcomabszz 39547 binomcxplemnotnn0 40694 limclner 41938 climinf2mpt 42001 climinfmpt 42002 dvnmptdivc 42229 dvnmul 42234 salpreimagtge 43009 salpreimaltle 43010 |
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