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Mirrors > Home > MPE Home > Th. List > Mathboxes > dropab1 | Structured version Visualization version GIF version |
Description: Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
dropab1 | ⊢ (∀𝑥 𝑥 = 𝑦 → {〈𝑥, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1 4795 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → 〈𝑥, 𝑧〉 = 〈𝑦, 𝑧〉) | |
2 | 1 | sps 2174 | . . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → 〈𝑥, 𝑧〉 = 〈𝑦, 𝑧〉) |
3 | 2 | eqeq2d 2829 | . . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 = 〈𝑥, 𝑧〉 ↔ 𝑤 = 〈𝑦, 𝑧〉)) |
4 | 3 | anbi1d 629 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑦, 𝑧〉 ∧ 𝜑))) |
5 | 4 | drex2 2456 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜑) ↔ ∃𝑧(𝑤 = 〈𝑦, 𝑧〉 ∧ 𝜑))) |
6 | 5 | drex1 2455 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜑) ↔ ∃𝑦∃𝑧(𝑤 = 〈𝑦, 𝑧〉 ∧ 𝜑))) |
7 | 6 | abbidv 2882 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑦∃𝑧(𝑤 = 〈𝑦, 𝑧〉 ∧ 𝜑)}) |
8 | df-opab 5120 | . 2 ⊢ {〈𝑥, 𝑧〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜑)} | |
9 | df-opab 5120 | . 2 ⊢ {〈𝑦, 𝑧〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑦∃𝑧(𝑤 = 〈𝑦, 𝑧〉 ∧ 𝜑)} | |
10 | 7, 8, 9 | 3eqtr4g 2878 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → {〈𝑥, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1526 = wceq 1528 ∃wex 1771 {cab 2796 〈cop 4563 {copab 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-13 2381 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-opab 5120 |
This theorem is referenced by: (None) |
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