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Mirrors > Home > MPE Home > Th. List > Mathboxes > prtlem5 | Structured version Visualization version GIF version |
Description: Lemma for prter1 34164, prter2 34166, prter3 34167 and prtex 34165. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
Ref | Expression |
---|---|
prtlem5 | ⊢ ([𝑠 / 𝑣][𝑟 / 𝑢]∃𝑥 ∈ 𝐴 (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝑟 ∈ 𝑥 ∧ 𝑠 ∈ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1843 | . 2 ⊢ Ⅎ𝑣∃𝑥 ∈ 𝐴 (𝑟 ∈ 𝑥 ∧ 𝑠 ∈ 𝑥) | |
2 | elequ1 1997 | . . . . 5 ⊢ (𝑢 = 𝑟 → (𝑢 ∈ 𝑥 ↔ 𝑟 ∈ 𝑥)) | |
3 | elequ1 1997 | . . . . 5 ⊢ (𝑣 = 𝑠 → (𝑣 ∈ 𝑥 ↔ 𝑠 ∈ 𝑥)) | |
4 | 2, 3 | bi2anan9r 918 | . . . 4 ⊢ ((𝑣 = 𝑠 ∧ 𝑢 = 𝑟) → ((𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ↔ (𝑟 ∈ 𝑥 ∧ 𝑠 ∈ 𝑥))) |
5 | 4 | rexbidv 3052 | . . 3 ⊢ ((𝑣 = 𝑠 ∧ 𝑢 = 𝑟) → (∃𝑥 ∈ 𝐴 (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝑟 ∈ 𝑥 ∧ 𝑠 ∈ 𝑥))) |
6 | 5 | sbiedv 2410 | . 2 ⊢ (𝑣 = 𝑠 → ([𝑟 / 𝑢]∃𝑥 ∈ 𝐴 (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝑟 ∈ 𝑥 ∧ 𝑠 ∈ 𝑥))) |
7 | 1, 6 | sbie 2408 | 1 ⊢ ([𝑠 / 𝑣][𝑟 / 𝑢]∃𝑥 ∈ 𝐴 (𝑢 ∈ 𝑥 ∧ 𝑣 ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐴 (𝑟 ∈ 𝑥 ∧ 𝑠 ∈ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 [wsb 1880 ∃wrex 2913 |
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-8 1992 ax-10 2019 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 df-sb 1881 df-rex 2918 |
This theorem is referenced by: (None) |
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