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Theorem copsex4g 3958
Description: An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.)
Hypothesis
Ref Expression
copsex4g.1 (((x = A y = B) (z = 𝐶 w = 𝐷)) → (φψ))
Assertion
Ref Expression
copsex4g (((A 𝑅 B 𝑆) (𝐶 𝑅 𝐷 𝑆)) → (xyzw((⟨A, B⟩ = ⟨x, y𝐶, 𝐷⟩ = ⟨z, w⟩) φ) ↔ ψ))
Distinct variable groups:   x,y,z,w,A   x,B,y,z,w   x,𝐶,y,z,w   x,𝐷,y,z,w   ψ,x,y,z,w   x,𝑅,y,z,w   x,𝑆,y,z,w
Allowed substitution hints:   φ(x,y,z,w)

Proof of Theorem copsex4g
StepHypRef Expression
1 eqcom 2024 . . . . . . 7 (⟨A, B⟩ = ⟨x, y⟩ ↔ ⟨x, y⟩ = ⟨A, B⟩)
2 vex 2538 . . . . . . . 8 x V
3 vex 2538 . . . . . . . 8 y V
42, 3opth 3948 . . . . . . 7 (⟨x, y⟩ = ⟨A, B⟩ ↔ (x = A y = B))
51, 4bitri 173 . . . . . 6 (⟨A, B⟩ = ⟨x, y⟩ ↔ (x = A y = B))
6 eqcom 2024 . . . . . . 7 (⟨𝐶, 𝐷⟩ = ⟨z, w⟩ ↔ ⟨z, w⟩ = ⟨𝐶, 𝐷⟩)
7 vex 2538 . . . . . . . 8 z V
8 vex 2538 . . . . . . . 8 w V
97, 8opth 3948 . . . . . . 7 (⟨z, w⟩ = ⟨𝐶, 𝐷⟩ ↔ (z = 𝐶 w = 𝐷))
106, 9bitri 173 . . . . . 6 (⟨𝐶, 𝐷⟩ = ⟨z, w⟩ ↔ (z = 𝐶 w = 𝐷))
115, 10anbi12i 436 . . . . 5 ((⟨A, B⟩ = ⟨x, y𝐶, 𝐷⟩ = ⟨z, w⟩) ↔ ((x = A y = B) (z = 𝐶 w = 𝐷)))
1211anbi1i 434 . . . 4 (((⟨A, B⟩ = ⟨x, y𝐶, 𝐷⟩ = ⟨z, w⟩) φ) ↔ (((x = A y = B) (z = 𝐶 w = 𝐷)) φ))
1312a1i 9 . . 3 (((A 𝑅 B 𝑆) (𝐶 𝑅 𝐷 𝑆)) → (((⟨A, B⟩ = ⟨x, y𝐶, 𝐷⟩ = ⟨z, w⟩) φ) ↔ (((x = A y = B) (z = 𝐶 w = 𝐷)) φ)))
14134exbidv 1732 . 2 (((A 𝑅 B 𝑆) (𝐶 𝑅 𝐷 𝑆)) → (xyzw((⟨A, B⟩ = ⟨x, y𝐶, 𝐷⟩ = ⟨z, w⟩) φ) ↔ xyzw(((x = A y = B) (z = 𝐶 w = 𝐷)) φ)))
15 id 19 . . 3 (((x = A y = B) (z = 𝐶 w = 𝐷)) → ((x = A y = B) (z = 𝐶 w = 𝐷)))
16 copsex4g.1 . . 3 (((x = A y = B) (z = 𝐶 w = 𝐷)) → (φψ))
1715, 16cgsex4g 2568 . 2 (((A 𝑅 B 𝑆) (𝐶 𝑅 𝐷 𝑆)) → (xyzw(((x = A y = B) (z = 𝐶 w = 𝐷)) φ) ↔ ψ))
1814, 17bitrd 177 1 (((A 𝑅 B 𝑆) (𝐶 𝑅 𝐷 𝑆)) → (xyzw((⟨A, B⟩ = ⟨x, y𝐶, 𝐷⟩ = ⟨z, w⟩) φ) ↔ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228  wex 1362   wcel 1374  cop 3353
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359
This theorem is referenced by:  opbrop  4346  ovi3  5560  dfplpq2  6213  dfmpq2  6214  enq0breq  6291
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