Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sngleq | Structured version Visualization version GIF version |
Description: Substitution property for sngl. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-sngleq | ⊢ (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexeq 3139 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑦 ∈ 𝐴 𝑥 = {𝑦} ↔ ∃𝑦 ∈ 𝐵 𝑥 = {𝑦})) | |
2 | 1 | abbidv 2741 | . 2 ⊢ (𝐴 = 𝐵 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}} = {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 = {𝑦}}) |
3 | df-bj-sngl 32954 | . 2 ⊢ sngl 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}} | |
4 | df-bj-sngl 32954 | . 2 ⊢ sngl 𝐵 = {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 = {𝑦}} | |
5 | 2, 3, 4 | 3eqtr4g 2681 | 1 ⊢ (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 {cab 2608 ∃wrex 2913 {csn 4177 sngl bj-csngl 32953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-bj-sngl 32954 |
This theorem is referenced by: bj-tageq 32964 |
Copyright terms: Public domain | W3C validator |