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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 2262. (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 3209 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝐴 |
4 | vtoclf.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | biimpd 219 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓)) |
6 | 3, 5 | eximii 1764 | . . 3 ⊢ ∃𝑥(𝜑 → 𝜓) |
7 | 1, 6 | 19.36i 2099 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) |
8 | vtoclf.4 | . 2 ⊢ 𝜑 | |
9 | 7, 8 | mpg 1724 | 1 ⊢ 𝜓 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 Vcvv 3200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-12 2047 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-v 3202 |
This theorem is referenced by: vtoclALT 3260 summolem2a 14446 prodmolem2a 14664 poimirlem24 33433 poimirlem28 33437 monotuz 37506 oddcomabszz 37509 binomcxplemnotnn0 38555 limclner 39883 climinf2mpt 39946 climinfmpt 39947 dvnmptdivc 40153 dvnmul 40158 salpreimagtge 40934 salpreimaltle 40935 |
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