Theorem swopolem 4105

Description: Perform the substitutions into the strict weak ordering law. (Contributed by Mario Carneiro, 31-Dec-2014.)

Hypothesis

Ref	Expression
swopolem.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))

Assertion

Ref	Expression
swopolem	⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍 ∨ 𝑍𝑅𝑌)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝜑,𝑥,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧   𝑦,𝑌,𝑧   𝑧,𝑍

Allowed substitution hints:   𝑌(𝑥)   𝑍(𝑥,𝑦)

Proof of Theorem swopolem

Step	Hyp	Ref	Expression
1		swopolem.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
2	1	ralrimivvva 2452	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
3		breq1 3823	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝑅𝑦 ↔ 𝑋𝑅𝑦))
4		breq1 3823	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥𝑅𝑧 ↔ 𝑋𝑅𝑧))
5	4	orbi1d 738	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦) ↔ (𝑋𝑅𝑧 ∨ 𝑧𝑅𝑦)))
6	3, 5	imbi12d 232	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) ↔ (𝑋𝑅𝑦 → (𝑋𝑅𝑧 ∨ 𝑧𝑅𝑦))))
7		breq2 3824	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝑅𝑦 ↔ 𝑋𝑅𝑌))
8		breq2 3824	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝑌))
9	8	orbi2d 737	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑋𝑅𝑧 ∨ 𝑧𝑅𝑦) ↔ (𝑋𝑅𝑧 ∨ 𝑧𝑅𝑌)))
10	7, 9	imbi12d 232	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋𝑅𝑦 → (𝑋𝑅𝑧 ∨ 𝑧𝑅𝑦)) ↔ (𝑋𝑅𝑌 → (𝑋𝑅𝑧 ∨ 𝑧𝑅𝑌))))
11		breq2 3824	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋𝑅𝑧 ↔ 𝑋𝑅𝑍))
12		breq1 3823	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧𝑅𝑌 ↔ 𝑍𝑅𝑌))
13	11, 12	orbi12d 740	. . . 4 ⊢ (𝑧 = 𝑍 → ((𝑋𝑅𝑧 ∨ 𝑧𝑅𝑌) ↔ (𝑋𝑅𝑍 ∨ 𝑍𝑅𝑌)))
14	13	imbi2d 228	. . 3 ⊢ (𝑧 = 𝑍 → ((𝑋𝑅𝑌 → (𝑋𝑅𝑧 ∨ 𝑧𝑅𝑌)) ↔ (𝑋𝑅𝑌 → (𝑋𝑅𝑍 ∨ 𝑍𝑅𝑌))))
15	6, 10, 14	rspc3v 2729	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍 ∨ 𝑍𝑅𝑌))))
16	2, 15	mpan9 275	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍 ∨ 𝑍𝑅𝑌)))

Syntax hints:   → wi 4   ∧ wa 102   ∨ wo 662   ∧ w3a 922   = wceq 1287   ∈ wcel 1436  ∀wral 2355   class class class wbr 3820

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2617  df-un 2992  df-sn 3437  df-pr 3438  df-op 3440  df-br 3821

This theorem is referenced by:  swoer  6266  swoord1  6267  swoord2  6268