Theorem swopolem 4266

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 2540	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
3		breq1 3969	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝑅𝑦 ↔ 𝑋𝑅𝑦))
4		breq1 3969	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥𝑅𝑧 ↔ 𝑋𝑅𝑧))
5	4	orbi1d 781	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦) ↔ (𝑋𝑅𝑧 ∨ 𝑧𝑅𝑦)))
6	3, 5	imbi12d 233	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) ↔ (𝑋𝑅𝑦 → (𝑋𝑅𝑧 ∨ 𝑧𝑅𝑦))))
7		breq2 3970	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝑅𝑦 ↔ 𝑋𝑅𝑌))
8		breq2 3970	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝑌))
9	8	orbi2d 780	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑋𝑅𝑧 ∨ 𝑧𝑅𝑦) ↔ (𝑋𝑅𝑧 ∨ 𝑧𝑅𝑌)))
10	7, 9	imbi12d 233	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋𝑅𝑦 → (𝑋𝑅𝑧 ∨ 𝑧𝑅𝑦)) ↔ (𝑋𝑅𝑌 → (𝑋𝑅𝑧 ∨ 𝑧𝑅𝑌))))
11		breq2 3970	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋𝑅𝑧 ↔ 𝑋𝑅𝑍))
12		breq1 3969	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧𝑅𝑌 ↔ 𝑍𝑅𝑌))
13	11, 12	orbi12d 783	. . . 4 ⊢ (𝑧 = 𝑍 → ((𝑋𝑅𝑧 ∨ 𝑧𝑅𝑌) ↔ (𝑋𝑅𝑍 ∨ 𝑍𝑅𝑌)))
14	13	imbi2d 229	. . 3 ⊢ (𝑧 = 𝑍 → ((𝑋𝑅𝑌 → (𝑋𝑅𝑧 ∨ 𝑧𝑅𝑌)) ↔ (𝑋𝑅𝑌 → (𝑋𝑅𝑍 ∨ 𝑍𝑅𝑌))))
15	6, 10, 14	rspc3v 2832	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍 ∨ 𝑍𝑅𝑌))))
16	2, 15	mpan9 279	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍 ∨ 𝑍𝑅𝑌)))

Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 698   ∧ w3a 963   = wceq 1335   ∈ wcel 2128  ∀wral 2435   class class class wbr 3966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-br 3967

This theorem is referenced by:  swoer  6509  swoord1  6510  swoord2  6511