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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1316 | 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 |
|---|---|
| bnj1316.1 | ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
| bnj1316.2 | ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) |
| Ref | Expression |
|---|---|
| bnj1316 | ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj1316.1 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | |
| 2 | 1 | nfcii 2894 | . . . 4 ⊢ Ⅎ𝑥𝐴 |
| 3 | bnj1316.2 | . . . . 5 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) | |
| 4 | 3 | nfcii 2894 | . . . 4 ⊢ Ⅎ𝑥𝐵 |
| 5 | 2, 4 | nfeq 2919 | . . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
| 6 | 5 | nf5ri 2195 | . 2 ⊢ (𝐴 = 𝐵 → ∀𝑥 𝐴 = 𝐵) |
| 7 | 6 | bnj956 34790 | 1 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ∪ 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rex 3071 df-iun 4993 |
| This theorem is referenced by: bnj1000 34955 bnj1318 35039 |
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