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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 2755 | . . . 4 ⊢ Ⅎ𝑥𝐴 |
3 | bnj1316.2 | . . . . 5 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) | |
4 | 3 | nfcii 2755 | . . . 4 ⊢ Ⅎ𝑥𝐵 |
5 | 2, 4 | nfeq 2776 | . . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
6 | 5 | nf5ri 2065 | . 2 ⊢ (𝐴 = 𝐵 → ∀𝑥 𝐴 = 𝐵) |
7 | 6 | bnj956 30847 | 1 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1481 = wceq 1483 ∈ wcel 1990 ∪ ciun 4520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-iun 4522 |
This theorem is referenced by: bnj1000 31011 bnj1318 31093 |
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