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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 2721 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
2 | 1 | anbi1i 624 | . . . 4 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓) ↔ (𝜑 ∧ 𝜓)) |
3 | 2 | exbii 1854 | . . 3 ⊢ (∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ 𝜓)) |
4 | 3 | abbii 2810 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)} = {𝑦 ∣ ∃𝑥(𝜑 ∧ 𝜓)} |
5 | nfab1 2911 | . . . . 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 216 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} → ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) |
9 | 5, 6, 8 | zfrepclf 5222 | . . . 4 ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)) |
10 | abeq2 2874 | . . . . 5 ⊢ (𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)} ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓))) | |
11 | 10 | exbii 1854 | . . . 4 ⊢ (∃𝑧 𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)} ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓))) |
12 | 9, 11 | mpbir 230 | . . 3 ⊢ ∃𝑧 𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)} |
13 | 12 | issetri 3447 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)} ∈ V |
14 | 4, 13 | eqeltrri 2838 | 1 ⊢ {𝑦 ∣ ∃𝑥(𝜑 ∧ 𝜓)} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1540 = wceq 1542 ∃wex 1786 ∈ wcel 2110 {cab 2717 Vcvv 3431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-v 3433 |
This theorem is referenced by: zfpair 5348 cshwsexa 14548 |
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