|   | Mathbox for Thierry Arnoux | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvdisjf | Structured version Visualization version GIF version | ||
| Description: Change bound variables in a disjoint collection. (Contributed by Thierry Arnoux, 6-Apr-2017.) | 
| Ref | Expression | 
|---|---|
| cbvdisjf.1 | ⊢ Ⅎ𝑥𝐴 | 
| cbvdisjf.2 | ⊢ Ⅎ𝑦𝐵 | 
| cbvdisjf.3 | ⊢ Ⅎ𝑥𝐶 | 
| cbvdisjf.4 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | 
| Ref | Expression | 
|---|---|
| cbvdisjf | ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfv 1914 | . . . . . 6 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 | |
| 2 | cbvdisjf.2 | . . . . . . 7 ⊢ Ⅎ𝑦𝐵 | |
| 3 | 2 | nfcri 2897 | . . . . . 6 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 | 
| 4 | 1, 3 | nfan 1899 | . . . . 5 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) | 
| 5 | cbvdisjf.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐴 | |
| 6 | 5 | nfcri 2897 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 | 
| 7 | cbvdisjf.3 | . . . . . . 7 ⊢ Ⅎ𝑥𝐶 | |
| 8 | 7 | nfcri 2897 | . . . . . 6 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶 | 
| 9 | 6, 8 | nfan 1899 | . . . . 5 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶) | 
| 10 | eleq1w 2824 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
| 11 | cbvdisjf.4 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
| 12 | 11 | eleq2d 2827 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶)) | 
| 13 | 10, 12 | anbi12d 632 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶))) | 
| 14 | 4, 9, 13 | cbvmow 2603 | . . . 4 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)) | 
| 15 | df-rmo 3380 | . . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) | |
| 16 | df-rmo 3380 | . . . 4 ⊢ (∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)) | |
| 17 | 14, 15, 16 | 3bitr4i 303 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) | 
| 18 | 17 | albii 1819 | . 2 ⊢ (∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) | 
| 19 | df-disj 5111 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵) | |
| 20 | df-disj 5111 | . 2 ⊢ (Disj 𝑦 ∈ 𝐴 𝐶 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) | |
| 21 | 18, 19, 20 | 3bitr4i 303 | 1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ∃*wmo 2538 Ⅎwnfc 2890 ∃*wrmo 3379 Disj wdisj 5110 | 
| 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-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-mo 2540 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rmo 3380 df-disj 5111 | 
| This theorem is referenced by: disjorsf 32593 ldgenpisyslem1 34164 | 
| Copyright terms: Public domain | W3C validator |