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Mirrors > Home > MPE Home > Th. List > vtoclef | Structured version Visualization version GIF version |
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.) |
Ref | Expression |
---|---|
vtoclef.1 | ⊢ Ⅎ𝑥𝜑 |
vtoclef.2 | ⊢ 𝐴 ∈ V |
vtoclef.3 | ⊢ (𝑥 = 𝐴 → 𝜑) |
Ref | Expression |
---|---|
vtoclef | ⊢ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtoclef.2 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | isseti 3427 | . 2 ⊢ ∃𝑥 𝑥 = 𝐴 |
3 | vtoclef.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
4 | vtoclef.3 | . . 3 ⊢ (𝑥 = 𝐴 → 𝜑) | |
5 | 3, 4 | exlimi 2262 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜑) |
6 | 2, 5 | ax-mp 5 | 1 ⊢ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∃wex 1880 Ⅎwnf 1884 ∈ wcel 2166 Vcvv 3415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-12 2222 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-v 3417 |
This theorem is referenced by: nn0ind-raph 11806 finxpreclem2 33773 |
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