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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-tageq | Structured version Visualization version GIF version |
Description: Substitution property for tag. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-tageq | ⊢ (𝐴 = 𝐵 → tag 𝐴 = tag 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-sngleq 34281 | . . 3 ⊢ (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵) | |
2 | 1 | uneq1d 4140 | . 2 ⊢ (𝐴 = 𝐵 → (sngl 𝐴 ∪ {∅}) = (sngl 𝐵 ∪ {∅})) |
3 | df-bj-tag 34289 | . 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅}) | |
4 | df-bj-tag 34289 | . 2 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅}) | |
5 | 2, 3, 4 | 3eqtr4g 2883 | 1 ⊢ (𝐴 = 𝐵 → tag 𝐴 = tag 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∪ cun 3936 ∅c0 4293 {csn 4569 sngl bj-csngl 34279 tag bj-ctag 34288 |
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 2795 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rex 3146 df-v 3498 df-un 3943 df-bj-sngl 34280 df-bj-tag 34289 |
This theorem is referenced by: bj-xtageq 34302 |
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