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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 32169 | . 2 ⊢ (𝐴 = 𝐵 → ({∅} × tag 𝐴) = ({∅} × tag 𝐵)) | |
2 | df-bj-1upl 32179 | . 2 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴) | |
3 | df-bj-1upl 32179 | . 2 ⊢ ⦅𝐵⦆ = ({∅} × tag 𝐵) | |
4 | 1, 2, 3 | 3eqtr4g 2669 | 1 ⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∅c0 3874 {csn 4125 × cxp 5036 tag bj-ctag 32155 ⦅bj-c1upl 32178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-v 3175 df-un 3545 df-opab 4644 df-xp 5044 df-bj-sngl 32147 df-bj-tag 32156 df-bj-1upl 32179 |
This theorem is referenced by: bj-1uplth 32188 bj-2upleq 32193 |
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