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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 36990 | . 2 ⊢ (𝐴 = 𝐵 → ({∅} × tag 𝐴) = ({∅} × tag 𝐵)) | |
| 2 | df-bj-1upl 37000 | . 2 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴) | |
| 3 | df-bj-1upl 37000 | . 2 ⊢ ⦅𝐵⦆ = ({∅} × tag 𝐵) | |
| 4 | 1, 2, 3 | 3eqtr4g 2801 | 1 ⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∅c0 4332 {csn 4625 × cxp 5682 tag bj-ctag 36976 ⦅bj-c1upl 36999 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rex 3070 df-v 3481 df-un 3955 df-opab 5205 df-xp 5690 df-bj-sngl 36968 df-bj-tag 36977 df-bj-1upl 37000 | 
| This theorem is referenced by: bj-1uplth 37009 bj-2upleq 37014 | 
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