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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 3636 | . 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 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 Vcvv 3474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 |
This theorem is referenced by: eqvincf 3638 dff13 7253 f1eqcocnv 7298 f1eqcocnvOLD 7299 tfindsg 7849 findsg 7889 findcard2s 9164 indpi 10901 fcoinvbr 31831 dfrdg4 34918 bj-elsngl 35844 |
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