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Mirrors > Home > MPE Home > Th. List > vtocld | Structured version Visualization version GIF version |
Description: Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-10 2130, ax-11 2147, ax-12 2167. (Revised by SN, 2-Sep-2024.) |
Ref | Expression |
---|---|
vtocld.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
vtocld.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
vtocld.3 | ⊢ (𝜑 → 𝜓) |
Ref | Expression |
---|---|
vtocld | ⊢ (𝜑 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtocld.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | elisset 2811 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴) |
4 | vtocld.3 | . . . 4 ⊢ (𝜑 → 𝜓) | |
5 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜓) |
6 | vtocld.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
7 | 5, 6 | mpbid 231 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝜒) |
8 | 3, 7 | exlimddv 1931 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-clel 2806 |
This theorem is referenced by: vtocl2d 3547 lmatfval 33415 lmatcl 33417 bj-elabd2ALT 36403 dvgrat 43749 dfatbrafv2b 46625 fnbrafv2b 46628 |
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