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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 2402. (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 3455 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝐴 |
4 | vtoclf.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | biimpd 232 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) |
6 | 3, 5 | eximii 1838 | . . 3 ⊢ ∃𝑥(𝜑 → 𝜓) |
7 | 1, 6 | 19.36i 2231 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
8 | vtoclf.4 | . 2 ⊢ 𝜑 | |
9 | 7, 8 | mpg 1799 | 1 ⊢ 𝜓 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 Vcvv 3441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-nf 1786 df-cleq 2791 df-clel 2870 |
This theorem is referenced by: vtoclALT 3508 summolem2a 15064 prodmolem2a 15280 poimirlem24 35081 poimirlem28 35085 monotuz 39882 oddcomabszz 39885 binomcxplemnotnn0 41060 limclner 42293 climinf2mpt 42356 climinfmpt 42357 dvnmptdivc 42580 dvnmul 42585 salpreimagtge 43359 salpreimaltle 43360 |
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