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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 4874 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → 〈𝑧, 𝑥〉 = 〈𝑧, 𝑦〉) | |
| 2 | 1 | sps 2185 | . . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → 〈𝑧, 𝑥〉 = 〈𝑧, 𝑦〉) |
| 3 | 2 | eqeq2d 2748 | . . . . . 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 2446 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑤 = 〈𝑧, 𝑥〉 ∧ 𝜑) ↔ ∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ 𝜑))) |
| 6 | 5 | drex2 2447 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧∃𝑥(𝑤 = 〈𝑧, 𝑥〉 ∧ 𝜑) ↔ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ 𝜑))) |
| 7 | 6 | abbidv 2808 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑤 ∣ ∃𝑧∃𝑥(𝑤 = 〈𝑧, 𝑥〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ 𝜑)}) |
| 8 | df-opab 5206 | . 2 ⊢ {〈𝑧, 𝑥〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑧∃𝑥(𝑤 = 〈𝑧, 𝑥〉 ∧ 𝜑)} | |
| 9 | df-opab 5206 | . 2 ⊢ {〈𝑧, 𝑦〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ 𝜑)} | |
| 10 | 7, 8, 9 | 3eqtr4g 2802 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → {〈𝑧, 𝑥〉 ∣ 𝜑} = {〈𝑧, 𝑦〉 ∣ 𝜑}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 ∃wex 1779 {cab 2714 〈cop 4632 {copab 5205 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 |
| This theorem is referenced by: (None) |
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