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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-1upleq | Structured version Visualization version GIF version |
Description: Substitution property for ⦅ − ⦆. (Contributed by BJ, 6-Apr-2019.) |
Ref | Expression |
---|---|
bj-1upleq | ⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-xtageq 32976 | . 2 ⊢ (𝐴 = 𝐵 → ({∅} × tag 𝐴) = ({∅} × tag 𝐵)) | |
2 | df-bj-1upl 32986 | . 2 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴) | |
3 | df-bj-1upl 32986 | . 2 ⊢ ⦅𝐵⦆ = ({∅} × tag 𝐵) | |
4 | 1, 2, 3 | 3eqtr4g 2681 | 1 ⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∅c0 3915 {csn 4177 × cxp 5112 tag bj-ctag 32962 ⦅bj-c1upl 32985 |
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-v 3202 df-un 3579 df-opab 4713 df-xp 5120 df-bj-sngl 32954 df-bj-tag 32963 df-bj-1upl 32986 |
This theorem is referenced by: bj-1uplth 32995 bj-2upleq 33000 |
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