Theorem swopolem 5044

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 2972	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
3		breq1 4656	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝑅𝑦 ↔ 𝑋𝑅𝑦))
4		breq1 4656	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥𝑅𝑧 ↔ 𝑋𝑅𝑧))
5	4	orbi1d 739	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦) ↔ (𝑋𝑅𝑧 ∨ 𝑧𝑅𝑦)))
6	3, 5	imbi12d 334	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) ↔ (𝑋𝑅𝑦 → (𝑋𝑅𝑧 ∨ 𝑧𝑅𝑦))))
7		breq2 4657	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝑅𝑦 ↔ 𝑋𝑅𝑌))
8		breq2 4657	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝑌))
9	8	orbi2d 738	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑋𝑅𝑧 ∨ 𝑧𝑅𝑦) ↔ (𝑋𝑅𝑧 ∨ 𝑧𝑅𝑌)))
10	7, 9	imbi12d 334	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋𝑅𝑦 → (𝑋𝑅𝑧 ∨ 𝑧𝑅𝑦)) ↔ (𝑋𝑅𝑌 → (𝑋𝑅𝑧 ∨ 𝑧𝑅𝑌))))
11		breq2 4657	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋𝑅𝑧 ↔ 𝑋𝑅𝑍))
12		breq1 4656	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧𝑅𝑌 ↔ 𝑍𝑅𝑌))
13	11, 12	orbi12d 746	. . . 4 ⊢ (𝑧 = 𝑍 → ((𝑋𝑅𝑧 ∨ 𝑧𝑅𝑌) ↔ (𝑋𝑅𝑍 ∨ 𝑍𝑅𝑌)))
14	13	imbi2d 330	. . 3 ⊢ (𝑧 = 𝑍 → ((𝑋𝑅𝑌 → (𝑋𝑅𝑧 ∨ 𝑧𝑅𝑌)) ↔ (𝑋𝑅𝑌 → (𝑋𝑅𝑍 ∨ 𝑍𝑅𝑌))))
15	6, 10, 14	rspc3v 3325	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍 ∨ 𝑍𝑅𝑌))))
16	2, 15	mpan9 486	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍 ∨ 𝑍𝑅𝑌)))

Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912   class class class wbr 4653

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-ral 2917  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-br 4654

This theorem is referenced by:  swoer  7772  swoord1  7773  swoord2  7774