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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 34283 | . . 3 ⊢ (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵) | |
2 | 1 | uneq1d 4141 | . 2 ⊢ (𝐴 = 𝐵 → (sngl 𝐴 ∪ {∅}) = (sngl 𝐵 ∪ {∅})) |
3 | df-bj-tag 34291 | . 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅}) | |
4 | df-bj-tag 34291 | . 2 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅}) | |
5 | 2, 3, 4 | 3eqtr4g 2884 | 1 ⊢ (𝐴 = 𝐵 → tag 𝐴 = tag 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∪ cun 3937 ∅c0 4294 {csn 4570 sngl bj-csngl 34281 tag bj-ctag 34290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rex 3147 df-v 3499 df-un 3944 df-bj-sngl 34282 df-bj-tag 34291 |
This theorem is referenced by: bj-xtageq 34304 |
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