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Mathbox for Andrew Salmon |
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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 4671 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → 〈𝑥, 𝑧〉 = 〈𝑦, 𝑧〉) | |
2 | 1 | sps 2113 | . . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → 〈𝑥, 𝑧〉 = 〈𝑦, 𝑧〉) |
3 | 2 | eqeq2d 2782 | . . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 = 〈𝑥, 𝑧〉 ↔ 𝑤 = 〈𝑦, 𝑧〉)) |
4 | 3 | anbi1d 620 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑦, 𝑧〉 ∧ 𝜑))) |
5 | 4 | drex2 2378 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜑) ↔ ∃𝑧(𝑤 = 〈𝑦, 𝑧〉 ∧ 𝜑))) |
6 | 5 | drex1 2377 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜑) ↔ ∃𝑦∃𝑧(𝑤 = 〈𝑦, 𝑧〉 ∧ 𝜑))) |
7 | 6 | abbidv 2837 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑦∃𝑧(𝑤 = 〈𝑦, 𝑧〉 ∧ 𝜑)}) |
8 | df-opab 4986 | . 2 ⊢ {〈𝑥, 𝑧〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑧(𝑤 = 〈𝑥, 𝑧〉 ∧ 𝜑)} | |
9 | df-opab 4986 | . 2 ⊢ {〈𝑦, 𝑧〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑦∃𝑧(𝑤 = 〈𝑦, 𝑧〉 ∧ 𝜑)} | |
10 | 7, 8, 9 | 3eqtr4g 2833 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → {〈𝑥, 𝑧〉 ∣ 𝜑} = {〈𝑦, 𝑧〉 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∀wal 1505 = wceq 1507 ∃wex 1742 {cab 2752 〈cop 4441 {copab 4985 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-rab 3091 df-v 3411 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-opab 4986 |
This theorem is referenced by: (None) |
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