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Theorem swopolem 5148
Description: Perform the substitutions into the strict weak ordering law. (Contributed by Mario Carneiro, 31-Dec-2014.)
Hypothesis
Ref Expression
swopolem.1 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
Assertion
Ref Expression
swopolem ((𝜑 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍𝑍𝑅𝑌)))
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
StepHypRef Expression
1 swopolem.1 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
21ralrimivvva 3074 . 2 (𝜑 → ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
3 breq1 4763 . . . 4 (𝑥 = 𝑋 → (𝑥𝑅𝑦𝑋𝑅𝑦))
4 breq1 4763 . . . . 5 (𝑥 = 𝑋 → (𝑥𝑅𝑧𝑋𝑅𝑧))
54orbi1d 741 . . . 4 (𝑥 = 𝑋 → ((𝑥𝑅𝑧𝑧𝑅𝑦) ↔ (𝑋𝑅𝑧𝑧𝑅𝑦)))
63, 5imbi12d 333 . . 3 (𝑥 = 𝑋 → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝑋𝑅𝑦 → (𝑋𝑅𝑧𝑧𝑅𝑦))))
7 breq2 4764 . . . 4 (𝑦 = 𝑌 → (𝑋𝑅𝑦𝑋𝑅𝑌))
8 breq2 4764 . . . . 5 (𝑦 = 𝑌 → (𝑧𝑅𝑦𝑧𝑅𝑌))
98orbi2d 740 . . . 4 (𝑦 = 𝑌 → ((𝑋𝑅𝑧𝑧𝑅𝑦) ↔ (𝑋𝑅𝑧𝑧𝑅𝑌)))
107, 9imbi12d 333 . . 3 (𝑦 = 𝑌 → ((𝑋𝑅𝑦 → (𝑋𝑅𝑧𝑧𝑅𝑦)) ↔ (𝑋𝑅𝑌 → (𝑋𝑅𝑧𝑧𝑅𝑌))))
11 breq2 4764 . . . . 5 (𝑧 = 𝑍 → (𝑋𝑅𝑧𝑋𝑅𝑍))
12 breq1 4763 . . . . 5 (𝑧 = 𝑍 → (𝑧𝑅𝑌𝑍𝑅𝑌))
1311, 12orbi12d 748 . . . 4 (𝑧 = 𝑍 → ((𝑋𝑅𝑧𝑧𝑅𝑌) ↔ (𝑋𝑅𝑍𝑍𝑅𝑌)))
1413imbi2d 329 . . 3 (𝑧 = 𝑍 → ((𝑋𝑅𝑌 → (𝑋𝑅𝑧𝑧𝑅𝑌)) ↔ (𝑋𝑅𝑌 → (𝑋𝑅𝑍𝑍𝑅𝑌))))
156, 10, 14rspc3v 3429 . 2 ((𝑋𝐴𝑌𝐴𝑍𝐴) → (∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍𝑍𝑅𝑌))))
162, 15mpan9 487 1 ((𝜑 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍𝑍𝑅𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383  w3a 1072   = wceq 1596  wcel 2103  wral 3014   class class class wbr 4760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761
This theorem is referenced by:  swoer  7892  swoord1  7893  swoord2  7894
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