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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 4403 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → 〈𝑧, 𝑥〉 = 〈𝑧, 𝑦〉) | |
2 | 1 | sps 2055 | . . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → 〈𝑧, 𝑥〉 = 〈𝑧, 𝑦〉) |
3 | 2 | eqeq2d 2632 | . . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 = 〈𝑧, 𝑥〉 ↔ 𝑤 = 〈𝑧, 𝑦〉)) |
4 | 3 | anbi1d 741 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → ((𝑤 = 〈𝑧, 𝑥〉 ∧ 𝜑) ↔ (𝑤 = 〈𝑧, 𝑦〉 ∧ 𝜑))) |
5 | 4 | drex1 2327 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑤 = 〈𝑧, 𝑥〉 ∧ 𝜑) ↔ ∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ 𝜑))) |
6 | 5 | drex2 2328 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧∃𝑥(𝑤 = 〈𝑧, 𝑥〉 ∧ 𝜑) ↔ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ 𝜑))) |
7 | 6 | abbidv 2741 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → {𝑤 ∣ ∃𝑧∃𝑥(𝑤 = 〈𝑧, 𝑥〉 ∧ 𝜑)} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ 𝜑)}) |
8 | df-opab 4713 | . 2 ⊢ {〈𝑧, 𝑥〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑧∃𝑥(𝑤 = 〈𝑧, 𝑥〉 ∧ 𝜑)} | |
9 | df-opab 4713 | . 2 ⊢ {〈𝑧, 𝑦〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑧∃𝑦(𝑤 = 〈𝑧, 𝑦〉 ∧ 𝜑)} | |
10 | 7, 8, 9 | 3eqtr4g 2681 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → {〈𝑧, 𝑥〉 ∣ 𝜑} = {〈𝑧, 𝑦〉 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∀wal 1481 = wceq 1483 ∃wex 1704 {cab 2608 〈cop 4183 {copab 4712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 |
This theorem is referenced by: (None) |
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