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Theorem swopolem 4033
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
StepHypRef Expression
1 swopolem.1 . . 3 ((φ (x A y A z A)) → (x𝑅y → (x𝑅z z𝑅y)))
21ralrimivvva 2396 . 2 (φx A y A z A (x𝑅y → (x𝑅z z𝑅y)))
3 breq1 3758 . . . 4 (x = 𝑋 → (x𝑅y𝑋𝑅y))
4 breq1 3758 . . . . 5 (x = 𝑋 → (x𝑅z𝑋𝑅z))
54orbi1d 704 . . . 4 (x = 𝑋 → ((x𝑅z z𝑅y) ↔ (𝑋𝑅z z𝑅y)))
63, 5imbi12d 223 . . 3 (x = 𝑋 → ((x𝑅y → (x𝑅z z𝑅y)) ↔ (𝑋𝑅y → (𝑋𝑅z z𝑅y))))
7 breq2 3759 . . . 4 (y = 𝑌 → (𝑋𝑅y𝑋𝑅𝑌))
8 breq2 3759 . . . . 5 (y = 𝑌 → (z𝑅yz𝑅𝑌))
98orbi2d 703 . . . 4 (y = 𝑌 → ((𝑋𝑅z z𝑅y) ↔ (𝑋𝑅z z𝑅𝑌)))
107, 9imbi12d 223 . . 3 (y = 𝑌 → ((𝑋𝑅y → (𝑋𝑅z z𝑅y)) ↔ (𝑋𝑅𝑌 → (𝑋𝑅z z𝑅𝑌))))
11 breq2 3759 . . . . 5 (z = 𝑍 → (𝑋𝑅z𝑋𝑅𝑍))
12 breq1 3758 . . . . 5 (z = 𝑍 → (z𝑅𝑌𝑍𝑅𝑌))
1311, 12orbi12d 706 . . . 4 (z = 𝑍 → ((𝑋𝑅z z𝑅𝑌) ↔ (𝑋𝑅𝑍 𝑍𝑅𝑌)))
1413imbi2d 219 . . 3 (z = 𝑍 → ((𝑋𝑅𝑌 → (𝑋𝑅z z𝑅𝑌)) ↔ (𝑋𝑅𝑌 → (𝑋𝑅𝑍 𝑍𝑅𝑌))))
156, 10, 14rspc3v 2659 . 2 ((𝑋 A 𝑌 A 𝑍 A) → (x A y A z A (x𝑅y → (x𝑅z z𝑅y)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍 𝑍𝑅𝑌))))
162, 15mpan9 265 1 ((φ (𝑋 A 𝑌 A 𝑍 A)) → (𝑋𝑅𝑌 → (𝑋𝑅𝑍 𝑍𝑅𝑌)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 628   w3a 884   = wceq 1242   wcel 1390  wral 2300   class class class wbr 3755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756
This theorem is referenced by:  swoer  6070  swoord1  6071  swoord2  6072
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