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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 2598 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
2 | 1 | anbi1i 727 | . . . 4 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓) ↔ (𝜑 ∧ 𝜓)) |
3 | 2 | exbii 1764 | . . 3 ⊢ (∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ 𝜓)) |
4 | 3 | abbii 2726 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)} = {𝑦 ∣ ∃𝑥(𝜑 ∧ 𝜓)} |
5 | nfab1 2753 | . . . . 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 206 | . . . . 5 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} → ∃𝑧∀𝑦(𝜓 → 𝑦 = 𝑧)) |
9 | 5, 6, 8 | zfrepclf 4705 | . . . 4 ⊢ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)) |
10 | abeq2 2719 | . . . . 5 ⊢ (𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)} ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓))) | |
11 | 10 | exbii 1764 | . . . 4 ⊢ (∃𝑧 𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)} ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓))) |
12 | 9, 11 | mpbir 220 | . . 3 ⊢ ∃𝑧 𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)} |
13 | 12 | issetri 3183 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ∧ 𝜓)} ∈ V |
14 | 4, 13 | eqeltrri 2685 | 1 ⊢ {𝑦 ∣ ∃𝑥(𝜑 ∧ 𝜓)} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {cab 2596 Vcvv 3173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 |
This theorem is referenced by: zfpair 4831 cshwsexa 13421 |
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