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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 2129, ax-11 2146, ax-12 2163. (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 2809 | . . 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 1930 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∃wex 1773 ∈ wcel 2098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-clel 2804 |
This theorem is referenced by: vtocl2d 3544 lmatfval 33324 lmatcl 33326 bj-elabd2ALT 36312 dvgrat 43628 dfatbrafv2b 46506 fnbrafv2b 46509 |
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