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Mirrors > Home > MPE Home > Th. List > zfrep4 | Structured version Visualization version GIF version |
Description: A version of Replacement using class abstractions. (Contributed by NM, 26-Nov-1995.) |
Ref | Expression |
---|---|
zfrep4.1 | ⊢ {𝑥 ∣ 𝜑} ∈ V |
zfrep4.2 | ⊢ (𝜑 → ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) |
Ref | Expression |
---|---|
zfrep4 | ⊢ {𝑦 ∣ ∃𝑥(𝜑 ∧ 𝜓)} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abid 2712 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
2 | 1 | anbi1i 733 | . . . 4 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓) ↔ (𝜑 ∧ 𝜓)) |
3 | 2 | exbii 1887 | . . 3 ⊢ (∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ 𝜓)) |
4 | 3 | abbii 2841 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)} = {𝑦 ∣ ∃𝑥(𝜑 ∧ 𝜓)} |
5 | nfab1 2868 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
6 | zfrep4.1 | . . . . 5 ⊢ {𝑥 ∣ 𝜑} ∈ V | |
7 | zfrep4.2 | . . . . . 6 ⊢ (𝜑 → ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) | |
8 | 1, 7 | sylbi 207 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} → ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) |
9 | 5, 6, 8 | zfrepclf 4885 | . . . 4 ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)) |
10 | abeq2 2834 | . . . . 5 ⊢ (𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)} ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓))) | |
11 | 10 | exbii 1887 | . . . 4 ⊢ (∃𝑧 𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)} ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓))) |
12 | 9, 11 | mpbir 221 | . . 3 ⊢ ∃𝑧 𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)} |
13 | 12 | issetri 3314 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)} ∈ V |
14 | 4, 13 | eqeltrri 2800 | 1 ⊢ {𝑦 ∣ ∃𝑥(𝜑 ∧ 𝜓)} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∀wal 1594 = wceq 1596 ∃wex 1817 ∈ wcel 2103 {cab 2710 Vcvv 3304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-13 2355 ax-ext 2704 ax-rep 4879 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1599 df-ex 1818 df-nf 1823 df-sb 2011 df-clab 2711 df-cleq 2717 df-clel 2720 df-nfc 2855 df-v 3306 |
This theorem is referenced by: zfpair 5009 cshwsexa 13691 |
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