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Mirrors > Home > MPE Home > Th. List > eqvinc | Structured version Visualization version GIF version |
Description: A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Thierry Arnoux, 23-Jan-2022.) |
Ref | Expression |
---|---|
eqvinc.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
eqvinc | ⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvinc.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | eqvincg 3479 | . 2 ⊢ (𝐴 ∈ V → (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 382 = wceq 1631 ∃wex 1852 ∈ wcel 2145 Vcvv 3351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-12 2203 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-tru 1634 df-ex 1853 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-v 3353 |
This theorem is referenced by: eqvincf 3481 dff13 6655 f1eqcocnv 6699 tfindsg 7207 findsg 7240 findcard2s 8357 indpi 9931 fcoinvbr 29757 dfrdg4 32395 bj-elsngl 33287 |
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