Theorem brswap2 4861

Description: Binary relationship equivalence for the Swap function. (Contributed by set.mm contributors, 8-Jan-2015.)

Hypotheses

Ref	Expression
br1st.1	⊢ B ∈ V
brswap.2	⊢ C ∈ V

Assertion

Ref	Expression
brswap2	⊢ (A Swap ⟨B, C⟩ ↔ A = ⟨C, B⟩)

Proof of Theorem brswap2

Dummy variables x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brex 4690	. . 3 ⊢ (A Swap ⟨B, C⟩ → (A ∈ V ∧ ⟨B, C⟩ ∈ V))
2	1	simpld 445	. 2 ⊢ (A Swap ⟨B, C⟩ → A ∈ V)
3		brswap.2	. . . 4 ⊢ C ∈ V
4		br1st.1	. . . 4 ⊢ B ∈ V
5	3, 4	opex 4589	. . 3 ⊢ ⟨C, B⟩ ∈ V
6		eleq1 2413	. . 3 ⊢ (A = ⟨C, B⟩ → (A ∈ V ↔ ⟨C, B⟩ ∈ V))
7	5, 6	mpbiri 224	. 2 ⊢ (A = ⟨C, B⟩ → A ∈ V)
8	4, 3	opex 4589	. . 3 ⊢ ⟨B, C⟩ ∈ V
9		eqeq1 2359	. . . . . 6 ⊢ (x = A → (x = ⟨z, w⟩ ↔ A = ⟨z, w⟩))
10	9	anbi1d 685	. . . . 5 ⊢ (x = A → ((x = ⟨z, w⟩ ∧ y = ⟨w, z⟩) ↔ (A = ⟨z, w⟩ ∧ y = ⟨w, z⟩)))
11	10	2exbidv 1628	. . . 4 ⊢ (x = A → (∃z∃w(x = ⟨z, w⟩ ∧ y = ⟨w, z⟩) ↔ ∃z∃w(A = ⟨z, w⟩ ∧ y = ⟨w, z⟩)))
12		eqeq1 2359	. . . . . . . . 9 ⊢ (y = ⟨B, C⟩ → (y = ⟨w, z⟩ ↔ ⟨B, C⟩ = ⟨w, z⟩))
13	12	anbi2d 684	. . . . . . . 8 ⊢ (y = ⟨B, C⟩ → ((A = ⟨z, w⟩ ∧ y = ⟨w, z⟩) ↔ (A = ⟨z, w⟩ ∧ ⟨B, C⟩ = ⟨w, z⟩)))
14		eqcom 2355	. . . . . . . . . . 11 ⊢ (⟨B, C⟩ = ⟨w, z⟩ ↔ ⟨w, z⟩ = ⟨B, C⟩)
15		opth 4603	. . . . . . . . . . 11 ⊢ (⟨w, z⟩ = ⟨B, C⟩ ↔ (w = B ∧ z = C))
16	14, 15	bitri 240	. . . . . . . . . 10 ⊢ (⟨B, C⟩ = ⟨w, z⟩ ↔ (w = B ∧ z = C))
17	16	anbi1i 676	. . . . . . . . 9 ⊢ ((⟨B, C⟩ = ⟨w, z⟩ ∧ A = ⟨z, w⟩) ↔ ((w = B ∧ z = C) ∧ A = ⟨z, w⟩))
18		ancom 437	. . . . . . . . 9 ⊢ ((A = ⟨z, w⟩ ∧ ⟨B, C⟩ = ⟨w, z⟩) ↔ (⟨B, C⟩ = ⟨w, z⟩ ∧ A = ⟨z, w⟩))
19		df-3an 936	. . . . . . . . 9 ⊢ ((w = B ∧ z = C ∧ A = ⟨z, w⟩) ↔ ((w = B ∧ z = C) ∧ A = ⟨z, w⟩))
20	17, 18, 19	3bitr4ri 269	. . . . . . . 8 ⊢ ((w = B ∧ z = C ∧ A = ⟨z, w⟩) ↔ (A = ⟨z, w⟩ ∧ ⟨B, C⟩ = ⟨w, z⟩))
21	13, 20	syl6bbr 254	. . . . . . 7 ⊢ (y = ⟨B, C⟩ → ((A = ⟨z, w⟩ ∧ y = ⟨w, z⟩) ↔ (w = B ∧ z = C ∧ A = ⟨z, w⟩)))
22	21	2exbidv 1628	. . . . . 6 ⊢ (y = ⟨B, C⟩ → (∃z∃w(A = ⟨z, w⟩ ∧ y = ⟨w, z⟩) ↔ ∃z∃w(w = B ∧ z = C ∧ A = ⟨z, w⟩)))
23		excom 1741	. . . . . 6 ⊢ (∃z∃w(w = B ∧ z = C ∧ A = ⟨z, w⟩) ↔ ∃w∃z(w = B ∧ z = C ∧ A = ⟨z, w⟩))
24	22, 23	syl6bb 252	. . . . 5 ⊢ (y = ⟨B, C⟩ → (∃z∃w(A = ⟨z, w⟩ ∧ y = ⟨w, z⟩) ↔ ∃w∃z(w = B ∧ z = C ∧ A = ⟨z, w⟩)))
25		opeq2 4580	. . . . . . 7 ⊢ (w = B → ⟨z, w⟩ = ⟨z, B⟩)
26	25	eqeq2d 2364	. . . . . 6 ⊢ (w = B → (A = ⟨z, w⟩ ↔ A = ⟨z, B⟩))
27		opeq1 4579	. . . . . . 7 ⊢ (z = C → ⟨z, B⟩ = ⟨C, B⟩)
28	27	eqeq2d 2364	. . . . . 6 ⊢ (z = C → (A = ⟨z, B⟩ ↔ A = ⟨C, B⟩))
29	4, 3, 26, 28	ceqsex2v 2897	. . . . 5 ⊢ (∃w∃z(w = B ∧ z = C ∧ A = ⟨z, w⟩) ↔ A = ⟨C, B⟩)
30	24, 29	syl6bb 252	. . . 4 ⊢ (y = ⟨B, C⟩ → (∃z∃w(A = ⟨z, w⟩ ∧ y = ⟨w, z⟩) ↔ A = ⟨C, B⟩))
31		df-swap 4725	. . . 4 ⊢ Swap = {⟨x, y⟩ ∣ ∃z∃w(x = ⟨z, w⟩ ∧ y = ⟨w, z⟩)}
32	11, 30, 31	brabg 4707	. . 3 ⊢ ((A ∈ V ∧ ⟨B, C⟩ ∈ V) → (A Swap ⟨B, C⟩ ↔ A = ⟨C, B⟩))
33	8, 32	mpan2 652	. 2 ⊢ (A ∈ V → (A Swap ⟨B, C⟩ ↔ A = ⟨C, B⟩))
34	2, 7, 33	pm5.21nii 342	1 ⊢ (A Swap ⟨B, C⟩ ↔ A = ⟨C, B⟩)

Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860  ⟨cop 4562   class class class wbr 4640   Swap cswap 4719

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  df-opab 4624  df-br 4641  df-swap 4725

This theorem is referenced by:  dfcnv2  5101  df2nd2  5112  swapf1o  5512  composeex  5821  domfnex  5871  connexex  5914  symex  5917