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Mathbox for Andrew Salmon |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dropab2 | 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 |
---|---|
dropab2 | ⊢ (∀𝑥 𝑥 = 𝑦 → {〈𝑧, 𝑥〉 ∣ 𝜑} = {〈𝑧, 𝑦〉 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq2 4879 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → 〈𝑧, 𝑥〉 = 〈𝑧, 𝑦〉) | |
2 | 1 | sps 2183 | . . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → 〈𝑧, 𝑥〉 = 〈𝑧, 𝑦〉) |
3 | 2 | eqeq2d 2746 | . . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 = 〈𝑧, 𝑥〉 ↔ 𝑤 = 〈𝑧, 𝑦〉)) |
4 | 3 | anbi1d 631 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑤 = 〈𝑧, 𝑥〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑧, 𝑦〉 ∧ 𝜑))) |
5 | 4 | drex1 2444 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑤 = 〈𝑧, 𝑥〉 ∧ 𝜑) ↔ ∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ 𝜑))) |
6 | 5 | drex2 2445 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧∃𝑥(𝑤 = 〈𝑧, 𝑥〉 ∧ 𝜑) ↔ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ 𝜑))) |
7 | 6 | abbidv 2806 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑤 ∣ ∃𝑧∃𝑥(𝑤 = 〈𝑧, 𝑥〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ 𝜑)}) |
8 | df-opab 5211 | . 2 ⊢ {〈𝑧, 𝑥〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑧∃𝑥(𝑤 = 〈𝑧, 𝑥〉 ∧ 𝜑)} | |
9 | df-opab 5211 | . 2 ⊢ {〈𝑧, 𝑦〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ 𝜑)} | |
10 | 7, 8, 9 | 3eqtr4g 2800 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → {〈𝑧, 𝑥〉 ∣ 𝜑} = {〈𝑧, 𝑦〉 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 = wceq 1537 ∃wex 1776 {cab 2712 〈cop 4637 {copab 5210 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-13 2375 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-opab 5211 |
This theorem is referenced by: (None) |
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