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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 34941 | . 2 ⊢ (𝐴 = 𝐵 → ({∅} × tag 𝐴) = ({∅} × tag 𝐵)) | |
2 | df-bj-1upl 34951 | . 2 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴) | |
3 | df-bj-1upl 34951 | . 2 ⊢ ⦅𝐵⦆ = ({∅} × tag 𝐵) | |
4 | 1, 2, 3 | 3eqtr4g 2804 | 1 ⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∅c0 4251 {csn 4555 × cxp 5563 tag bj-ctag 34927 ⦅bj-c1upl 34950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-ext 2709 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-ral 3067 df-rex 3068 df-v 3422 df-un 3885 df-opab 5130 df-xp 5571 df-bj-sngl 34919 df-bj-tag 34928 df-bj-1upl 34951 |
This theorem is referenced by: bj-1uplth 34960 bj-2upleq 34965 |
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