Theorem copsex4g 4611

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)) → (φ ↔ ψ))

Assertion

Ref	Expression
copsex4g	⊢ (((A ∈ R ∧ B ∈ S) ∧ (C ∈ R ∧ D ∈ S)) → (∃x∃y∃z∃w((⟨A, B⟩ = ⟨x, y⟩ ∧ ⟨C, 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

Step	Hyp	Ref	Expression
1		eqcom 2355	. . . . . . 7 ⊢ (⟨A, B⟩ = ⟨x, y⟩ ↔ ⟨x, y⟩ = ⟨A, B⟩)
2		opth 4603	. . . . . . 7 ⊢ (⟨x, y⟩ = ⟨A, B⟩ ↔ (x = A ∧ y = B))
3	1, 2	bitri 240	. . . . . 6 ⊢ (⟨A, B⟩ = ⟨x, y⟩ ↔ (x = A ∧ y = B))
4		eqcom 2355	. . . . . . 7 ⊢ (⟨C, D⟩ = ⟨z, w⟩ ↔ ⟨z, w⟩ = ⟨C, D⟩)
5		opth 4603	. . . . . . 7 ⊢ (⟨z, w⟩ = ⟨C, D⟩ ↔ (z = C ∧ w = D))
6	4, 5	bitri 240	. . . . . 6 ⊢ (⟨C, D⟩ = ⟨z, w⟩ ↔ (z = C ∧ w = D))
7	3, 6	anbi12i 678	. . . . 5 ⊢ ((⟨A, B⟩ = ⟨x, y⟩ ∧ ⟨C, D⟩ = ⟨z, w⟩) ↔ ((x = A ∧ y = B) ∧ (z = C ∧ w = D)))
8	7	anbi1i 676	. . . 4 ⊢ (((⟨A, B⟩ = ⟨x, y⟩ ∧ ⟨C, D⟩ = ⟨z, w⟩) ∧ φ) ↔ (((x = A ∧ y = B) ∧ (z = C ∧ w = D)) ∧ φ))
9	8	2exbii 1583	. . 3 ⊢ (∃z∃w((⟨A, B⟩ = ⟨x, y⟩ ∧ ⟨C, D⟩ = ⟨z, w⟩) ∧ φ) ↔ ∃z∃w(((x = A ∧ y = B) ∧ (z = C ∧ w = D)) ∧ φ))
10	9	2exbii 1583	. 2 ⊢ (∃x∃y∃z∃w((⟨A, B⟩ = ⟨x, y⟩ ∧ ⟨C, D⟩ = ⟨z, w⟩) ∧ φ) ↔ ∃x∃y∃z∃w(((x = A ∧ y = B) ∧ (z = C ∧ w = D)) ∧ φ))
11		id 19	. . 3 ⊢ (((x = A ∧ y = B) ∧ (z = C ∧ w = D)) → ((x = A ∧ y = B) ∧ (z = C ∧ w = D)))
12		copsex4g.1	. . 3 ⊢ (((x = A ∧ y = B) ∧ (z = C ∧ w = D)) → (φ ↔ ψ))
13	11, 12	cgsex4g 2893	. 2 ⊢ (((A ∈ R ∧ B ∈ S) ∧ (C ∈ R ∧ D ∈ S)) → (∃x∃y∃z∃w(((x = A ∧ y = B) ∧ (z = C ∧ w = D)) ∧ φ) ↔ ψ))
14	10, 13	syl5bb 248	1 ⊢ (((A ∈ R ∧ B ∈ S) ∧ (C ∈ R ∧ D ∈ S)) → (∃x∃y∃z∃w((⟨A, B⟩ = ⟨x, y⟩ ∧ ⟨C, D⟩ = ⟨z, w⟩) ∧ φ) ↔ ψ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ⟨cop 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569

This theorem is referenced by:  opbrop  4842  ov3  5600