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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1113 | Structured version Visualization version GIF version | ||
| Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| bnj1113.1 | ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐷) |
| Ref | Expression |
|---|---|
| bnj1113 | ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐶 𝐸 = ∪ 𝑥 ∈ 𝐷 𝐸) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj1113.1 | . 2 ⊢ (𝐴 = 𝐵 → 𝐶 = 𝐷) | |
| 2 | 1 | iuneq1d 5019 | 1 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐶 𝐸 = ∪ 𝑥 ∈ 𝐷 𝐸) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∪ ciun 4991 |
| 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-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-v 3482 df-ss 3968 df-iun 4993 |
| This theorem is referenced by: bnj106 34882 bnj222 34897 bnj540 34906 bnj553 34912 bnj611 34932 bnj966 34958 bnj1112 34997 |
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