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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 36968 | . . 3 ⊢ (𝐴 = 𝐵 → sngl 𝐴 = sngl 𝐵) | |
| 2 | 1 | uneq1d 4167 | . 2 ⊢ (𝐴 = 𝐵 → (sngl 𝐴 ∪ {∅}) = (sngl 𝐵 ∪ {∅})) |
| 3 | df-bj-tag 36976 | . 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅}) | |
| 4 | df-bj-tag 36976 | . 2 ⊢ tag 𝐵 = (sngl 𝐵 ∪ {∅}) | |
| 5 | 2, 3, 4 | 3eqtr4g 2802 | 1 ⊢ (𝐴 = 𝐵 → tag 𝐴 = tag 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∪ cun 3949 ∅c0 4333 {csn 4626 sngl bj-csngl 36966 tag bj-ctag 36975 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-v 3482 df-un 3956 df-bj-sngl 36967 df-bj-tag 36976 |
| This theorem is referenced by: bj-xtageq 36989 |
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