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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege54cor1c | Structured version Visualization version GIF version |
Description: Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Revised by RP, 25-Apr-2020.) |
Ref | Expression |
---|---|
frege54c.1 | ⊢ 𝐴 ∈ 𝐶 |
Ref | Expression |
---|---|
frege54cor1c | ⊢ [𝐴 / 𝑥]𝑥 = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege54c.1 | . . . . 5 ⊢ 𝐴 ∈ 𝐶 | |
2 | 1 | elexi 3365 | . . . 4 ⊢ 𝐴 ∈ V |
3 | 2 | snid 4348 | . . 3 ⊢ 𝐴 ∈ {𝐴} |
4 | df-sn 4318 | . . 3 ⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴} | |
5 | 3, 4 | eleqtri 2848 | . 2 ⊢ 𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐴} |
6 | df-sbc 3589 | . 2 ⊢ ([𝐴 / 𝑥]𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐴}) | |
7 | 5, 6 | mpbir 221 | 1 ⊢ [𝐴 / 𝑥]𝑥 = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1631 ∈ wcel 2145 {cab 2757 [wsbc 3588 {csn 4317 |
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-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-v 3353 df-sbc 3589 df-sn 4318 |
This theorem is referenced by: frege55lem2c 38738 frege55c 38739 frege56c 38740 |
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