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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 3638 | . 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 207 ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 Vcvv 3492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-cleq 2811 df-clel 2890 |
This theorem is referenced by: eqvincf 3640 dff13 7004 f1eqcocnv 7048 tfindsg 7564 findsg 7598 findcard2s 8747 indpi 10317 fcoinvbr 30286 dfrdg4 33309 bj-elsngl 34177 |
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