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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-2upleq | Structured version Visualization version GIF version |
Description: Substitution property for ⦅ − , − ⦆. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-2upleq | ⊢ (𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-1upleq 34314 | . . 3 ⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆) | |
2 | bj-xtageq 34303 | . . 3 ⊢ (𝐶 = 𝐷 → ({1o} × tag 𝐶) = ({1o} × tag 𝐷)) | |
3 | uneq12 4134 | . . . 4 ⊢ ((⦅𝐴⦆ = ⦅𝐵⦆ ∧ ({1o} × tag 𝐶) = ({1o} × tag 𝐷)) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷))) | |
4 | 3 | ex 415 | . . 3 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ → (({1o} × tag 𝐶) = ({1o} × tag 𝐷) → (⦅𝐴⦆ ∪ ({1o} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷)))) |
5 | 1, 2, 4 | syl2im 40 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 = 𝐷 → (⦅𝐴⦆ ∪ ({1o} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷)))) |
6 | df-bj-2upl 34326 | . . 3 ⊢ ⦅𝐴, 𝐶⦆ = (⦅𝐴⦆ ∪ ({1o} × tag 𝐶)) | |
7 | df-bj-2upl 34326 | . . 3 ⊢ ⦅𝐵, 𝐷⦆ = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷)) | |
8 | 6, 7 | eqeq12i 2836 | . 2 ⊢ (⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆ ↔ (⦅𝐴⦆ ∪ ({1o} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1o} × tag 𝐷))) |
9 | 5, 8 | syl6ibr 254 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∪ cun 3934 {csn 4567 × cxp 5553 1oc1o 8095 tag bj-ctag 34289 ⦅bj-c1upl 34312 ⦅bj-c2uple 34325 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-v 3496 df-un 3941 df-opab 5129 df-xp 5561 df-bj-sngl 34281 df-bj-tag 34290 df-bj-1upl 34313 df-bj-2upl 34326 |
This theorem is referenced by: bj-2uplth 34336 |
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