Theorem copsex2t 4609

Description: Closed theorem form of copsex2g 4610. (Contributed by NM, 17-Feb-2013.)

Assertion

Ref	Expression
copsex2t	⊢ ((∀x∀y((x = A ∧ y = B) → (φ ↔ ψ)) ∧ (A ∈ V ∧ B ∈ W)) → (∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) ↔ ψ))

Distinct variable groups:   x,y,ψ   x,A,y   x,B,y

Allowed substitution hints:   φ(x,y)   V(x,y)   W(x,y)

Proof of Theorem copsex2t

Step	Hyp	Ref	Expression
1		elisset 2870	. . . 4 ⊢ (A ∈ V → ∃x x = A)
2		elisset 2870	. . . 4 ⊢ (B ∈ W → ∃y y = B)
3	1, 2	anim12i 549	. . 3 ⊢ ((A ∈ V ∧ B ∈ W) → (∃x x = A ∧ ∃y y = B))
4		eeanv 1913	. . 3 ⊢ (∃x∃y(x = A ∧ y = B) ↔ (∃x x = A ∧ ∃y y = B))
5	3, 4	sylibr 203	. 2 ⊢ ((A ∈ V ∧ B ∈ W) → ∃x∃y(x = A ∧ y = B))
6		nfa1 1788	. . . 4 ⊢ Ⅎx∀x∀y((x = A ∧ y = B) → (φ ↔ ψ))
7		nfe1 1732	. . . . 5 ⊢ Ⅎx∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ)
8		nfv 1619	. . . . 5 ⊢ Ⅎxψ
9	7, 8	nfbi 1834	. . . 4 ⊢ Ⅎx(∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) ↔ ψ)
10		nfa2 1855	. . . . 5 ⊢ Ⅎy∀x∀y((x = A ∧ y = B) → (φ ↔ ψ))
11		nfe1 1732	. . . . . . 7 ⊢ Ⅎy∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ)
12	11	nfex 1843	. . . . . 6 ⊢ Ⅎy∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ)
13		nfv 1619	. . . . . 6 ⊢ Ⅎyψ
14	12, 13	nfbi 1834	. . . . 5 ⊢ Ⅎy(∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) ↔ ψ)
15		opeq12 4581	. . . . . . . . 9 ⊢ ((x = A ∧ y = B) → ⟨x, y⟩ = ⟨A, B⟩)
16		copsexg 4608	. . . . . . . . . 10 ⊢ (⟨A, B⟩ = ⟨x, y⟩ → (φ ↔ ∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ)))
17	16	eqcoms 2356	. . . . . . . . 9 ⊢ (⟨x, y⟩ = ⟨A, B⟩ → (φ ↔ ∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ)))
18	15, 17	syl 15	. . . . . . . 8 ⊢ ((x = A ∧ y = B) → (φ ↔ ∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ)))
19	18	adantl 452	. . . . . . 7 ⊢ ((∀x∀y((x = A ∧ y = B) → (φ ↔ ψ)) ∧ (x = A ∧ y = B)) → (φ ↔ ∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ)))
20		sp 1747	. . . . . . . . 9 ⊢ (∀x∀y((x = A ∧ y = B) → (φ ↔ ψ)) → ∀y((x = A ∧ y = B) → (φ ↔ ψ)))
21	20	19.21bi 1758	. . . . . . . 8 ⊢ (∀x∀y((x = A ∧ y = B) → (φ ↔ ψ)) → ((x = A ∧ y = B) → (φ ↔ ψ)))
22	21	imp 418	. . . . . . 7 ⊢ ((∀x∀y((x = A ∧ y = B) → (φ ↔ ψ)) ∧ (x = A ∧ y = B)) → (φ ↔ ψ))
23	19, 22	bitr3d 246	. . . . . 6 ⊢ ((∀x∀y((x = A ∧ y = B) → (φ ↔ ψ)) ∧ (x = A ∧ y = B)) → (∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) ↔ ψ))
24	23	ex 423	. . . . 5 ⊢ (∀x∀y((x = A ∧ y = B) → (φ ↔ ψ)) → ((x = A ∧ y = B) → (∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) ↔ ψ)))
25	10, 14, 24	exlimd 1806	. . . 4 ⊢ (∀x∀y((x = A ∧ y = B) → (φ ↔ ψ)) → (∃y(x = A ∧ y = B) → (∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) ↔ ψ)))
26	6, 9, 25	exlimd 1806	. . 3 ⊢ (∀x∀y((x = A ∧ y = B) → (φ ↔ ψ)) → (∃x∃y(x = A ∧ y = B) → (∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) ↔ ψ)))
27	26	imp 418	. 2 ⊢ ((∀x∀y((x = A ∧ y = B) → (φ ↔ ψ)) ∧ ∃x∃y(x = A ∧ y = B)) → (∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) ↔ ψ))
28	5, 27	sylan2 460	1 ⊢ ((∀x∀y((x = A ∧ y = B) → (φ ↔ ψ)) ∧ (A ∈ V ∧ B ∈ W)) → (∃x∃y(⟨A, B⟩ = ⟨x, y⟩ ∧ φ) ↔ ψ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃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:  opelopabt  4700