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Mirrors > Home > MPE Home > Th. List > zfrepclf | Structured version Visualization version GIF version |
Description: An inference based on the Axiom of Replacement. Typically, 𝜑 defines a function from 𝑥 to 𝑦. (Contributed by NM, 26-Nov-1995.) |
Ref | Expression |
---|---|
zfrepclf.1 | ⊢ Ⅎ𝑥𝐴 |
zfrepclf.2 | ⊢ 𝐴 ∈ V |
zfrepclf.3 | ⊢ (𝑥 ∈ 𝐴 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧)) |
Ref | Expression |
---|---|
zfrepclf | ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zfrepclf.2 | . 2 ⊢ 𝐴 ∈ V | |
2 | zfrepclf.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfeq2 2914 | . . . . 5 ⊢ Ⅎ𝑥 𝑣 = 𝐴 |
4 | eleq2 2816 | . . . . . 6 ⊢ (𝑣 = 𝐴 → (𝑥 ∈ 𝑣 ↔ 𝑥 ∈ 𝐴)) | |
5 | zfrepclf.3 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧)) | |
6 | 4, 5 | biimtrdi 252 | . . . . 5 ⊢ (𝑣 = 𝐴 → (𝑥 ∈ 𝑣 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧))) |
7 | 3, 6 | alrimi 2198 | . . . 4 ⊢ (𝑣 = 𝐴 → ∀𝑥(𝑥 ∈ 𝑣 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧))) |
8 | nfv 1909 | . . . . 5 ⊢ Ⅎ𝑧𝜑 | |
9 | 8 | axrep5 5284 | . . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑣 → ∃𝑧∀𝑦(𝜑 → 𝑦 = 𝑧)) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑣 ∧ 𝜑))) |
10 | 7, 9 | syl 17 | . . 3 ⊢ (𝑣 = 𝐴 → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑣 ∧ 𝜑))) |
11 | 4 | anbi1d 629 | . . . . . . 7 ⊢ (𝑣 = 𝐴 → ((𝑥 ∈ 𝑣 ∧ 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))) |
12 | 3, 11 | exbid 2208 | . . . . . 6 ⊢ (𝑣 = 𝐴 → (∃𝑥(𝑥 ∈ 𝑣 ∧ 𝜑) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) |
13 | 12 | bibi2d 342 | . . . . 5 ⊢ (𝑣 = 𝐴 → ((𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑣 ∧ 𝜑)) ↔ (𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))) |
14 | 13 | albidv 1915 | . . . 4 ⊢ (𝑣 = 𝐴 → (∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑣 ∧ 𝜑)) ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))) |
15 | 14 | exbidv 1916 | . . 3 ⊢ (𝑣 = 𝐴 → (∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑣 ∧ 𝜑)) ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))) |
16 | 10, 15 | mpbid 231 | . 2 ⊢ (𝑣 = 𝐴 → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) |
17 | 1, 16 | vtocle 3538 | 1 ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1531 = wceq 1533 ∃wex 1773 ∈ wcel 2098 Ⅎwnfc 2877 Vcvv 3468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 |
This theorem is referenced by: zfrep3cl 5288 zfrep4 5289 |
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