![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > vtoclgOLD | Structured version Visualization version GIF version |
Description: Obsolete version of vtoclg 3527 as of 26-Jan-2025. (Contributed by NM, 17-Apr-1995.) Avoid ax-12 2172. (Revised by SN, 20-Apr-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
vtoclg.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
vtoclg.2 | ⊢ 𝜑 |
Ref | Expression |
---|---|
vtoclgOLD | ⊢ (𝐴 ∈ 𝑉 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elisset 2816 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) | |
2 | vtoclg.2 | . . . 4 ⊢ 𝜑 | |
3 | vtoclg.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | mpbii 232 | . . 3 ⊢ (𝑥 = 𝐴 → 𝜓) |
5 | 4 | exlimiv 1934 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 → 𝜓) |
6 | 1, 5 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∃wex 1782 ∈ wcel 2107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-clel 2811 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |