![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
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 3415 | . . . 4 ⊢ 𝐴 ∈ V |
3 | 2 | snid 4430 | . . 3 ⊢ 𝐴 ∈ {𝐴} |
4 | df-sn 4399 | . . 3 ⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴} | |
5 | 3, 4 | eleqtri 2857 | . 2 ⊢ 𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐴} |
6 | df-sbc 3653 | . 2 ⊢ ([𝐴 / 𝑥]𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐴}) | |
7 | 5, 6 | mpbir 223 | 1 ⊢ [𝐴 / 𝑥]𝑥 = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2107 {cab 2763 [wsbc 3652 {csn 4398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-v 3400 df-sbc 3653 df-sn 4399 |
This theorem is referenced by: frege55lem2c 39167 frege55c 39168 frege56c 39169 |
Copyright terms: Public domain | W3C validator |