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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 3589 | . 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 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 Vcvv 3441 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-cleq 2791 df-clel 2870 |
This theorem is referenced by: eqvincf 3591 dff13 6991 f1eqcocnv 7035 f1eqcocnvOLD 7036 tfindsg 7555 findsg 7590 findcard2s 8743 indpi 10318 fcoinvbr 30371 dfrdg4 33525 bj-elsngl 34404 |
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