Theorem opelopabsb 4698

Description: The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)

Assertion

Ref	Expression
opelopabsb	⊢ (⟨A, B⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ [̣A / x]̣[̣B / y]̣φ)

Distinct variable groups:   x,y   x,B

Allowed substitution hints:   φ(x,y)   A(x,y)   B(y)

Proof of Theorem opelopabsb

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

Step	Hyp	Ref	Expression
1		df-br 4641	. . 3 ⊢ (A{⟨x, y⟩ ∣ φ}B ↔ ⟨A, B⟩ ∈ {⟨x, y⟩ ∣ φ})
2		brex 4690	. . 3 ⊢ (A{⟨x, y⟩ ∣ φ}B → (A ∈ V ∧ B ∈ V))
3	1, 2	sylbir 204	. 2 ⊢ (⟨A, B⟩ ∈ {⟨x, y⟩ ∣ φ} → (A ∈ V ∧ B ∈ V))
4		sbcex 3056	. . 3 ⊢ ([̣A / x]̣[̣B / y]̣φ → A ∈ V)
5		spesbc 3128	. . . 4 ⊢ ([̣A / x]̣[̣B / y]̣φ → ∃x[̣B / y]̣φ)
6		sbcex 3056	. . . . 5 ⊢ ([̣B / y]̣φ → B ∈ V)
7	6	exlimiv 1634	. . . 4 ⊢ (∃x[̣B / y]̣φ → B ∈ V)
8	5, 7	syl 15	. . 3 ⊢ ([̣A / x]̣[̣B / y]̣φ → B ∈ V)
9	4, 8	jca 518	. 2 ⊢ ([̣A / x]̣[̣B / y]̣φ → (A ∈ V ∧ B ∈ V))
10		opeq1 4579	. . . . 5 ⊢ (z = A → ⟨z, w⟩ = ⟨A, w⟩)
11	10	eleq1d 2419	. . . 4 ⊢ (z = A → (⟨z, w⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ ⟨A, w⟩ ∈ {⟨x, y⟩ ∣ φ}))
12		dfsbcq2 3050	. . . 4 ⊢ (z = A → ([z / x][w / y]φ ↔ [̣A / x]̣[w / y]φ))
13	11, 12	bibi12d 312	. . 3 ⊢ (z = A → ((⟨z, w⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ [z / x][w / y]φ) ↔ (⟨A, w⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ [̣A / x]̣[w / y]φ)))
14		opeq2 4580	. . . . 5 ⊢ (w = B → ⟨A, w⟩ = ⟨A, B⟩)
15	14	eleq1d 2419	. . . 4 ⊢ (w = B → (⟨A, w⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ ⟨A, B⟩ ∈ {⟨x, y⟩ ∣ φ}))
16		dfsbcq2 3050	. . . . 5 ⊢ (w = B → ([w / y]φ ↔ [̣B / y]̣φ))
17	16	sbcbidv 3101	. . . 4 ⊢ (w = B → ([̣A / x]̣[w / y]φ ↔ [̣A / x]̣[̣B / y]̣φ))
18	15, 17	bibi12d 312	. . 3 ⊢ (w = B → ((⟨A, w⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ [̣A / x]̣[w / y]φ) ↔ (⟨A, B⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ [̣A / x]̣[̣B / y]̣φ)))
19		nfopab1 4629	. . . . . 6 ⊢ Ⅎx{⟨x, y⟩ ∣ φ}
20	19	nfel2 2502	. . . . 5 ⊢ Ⅎx⟨z, w⟩ ∈ {⟨x, y⟩ ∣ φ}
21		nfs1v 2106	. . . . 5 ⊢ Ⅎx[z / x][w / y]φ
22	20, 21	nfbi 1834	. . . 4 ⊢ Ⅎx(⟨z, w⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ [z / x][w / y]φ)
23		opeq1 4579	. . . . . 6 ⊢ (x = z → ⟨x, w⟩ = ⟨z, w⟩)
24	23	eleq1d 2419	. . . . 5 ⊢ (x = z → (⟨x, w⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ ⟨z, w⟩ ∈ {⟨x, y⟩ ∣ φ}))
25		sbequ12 1919	. . . . 5 ⊢ (x = z → ([w / y]φ ↔ [z / x][w / y]φ))
26	24, 25	bibi12d 312	. . . 4 ⊢ (x = z → ((⟨x, w⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ [w / y]φ) ↔ (⟨z, w⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ [z / x][w / y]φ)))
27		nfopab2 4630	. . . . . . 7 ⊢ Ⅎy{⟨x, y⟩ ∣ φ}
28	27	nfel2 2502	. . . . . 6 ⊢ Ⅎy⟨x, w⟩ ∈ {⟨x, y⟩ ∣ φ}
29		nfs1v 2106	. . . . . 6 ⊢ Ⅎy[w / y]φ
30	28, 29	nfbi 1834	. . . . 5 ⊢ Ⅎy(⟨x, w⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ [w / y]φ)
31		opeq2 4580	. . . . . . 7 ⊢ (y = w → ⟨x, y⟩ = ⟨x, w⟩)
32	31	eleq1d 2419	. . . . . 6 ⊢ (y = w → (⟨x, y⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ ⟨x, w⟩ ∈ {⟨x, y⟩ ∣ φ}))
33		sbequ12 1919	. . . . . 6 ⊢ (y = w → (φ ↔ [w / y]φ))
34	32, 33	bibi12d 312	. . . . 5 ⊢ (y = w → ((⟨x, y⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ φ) ↔ (⟨x, w⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ [w / y]φ)))
35		opabid 4696	. . . . 5 ⊢ (⟨x, y⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ φ)
36	30, 34, 35	chvar 1986	. . . 4 ⊢ (⟨x, w⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ [w / y]φ)
37	22, 26, 36	chvar 1986	. . 3 ⊢ (⟨z, w⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ [z / x][w / y]φ)
38	13, 18, 37	vtocl2g 2919	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (⟨A, B⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ [̣A / x]̣[̣B / y]̣φ))
39	3, 9, 38	pm5.21nii 342	1 ⊢ (⟨A, B⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ [̣A / x]̣[̣B / y]̣φ)

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642  [wsb 1648   ∈ wcel 1710  Vcvv 2860  [̣wsbc 3047  ⟨cop 4562  {copab 4623   class class class wbr 4640

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

This theorem is referenced by:  brabsb  4699  opelopabaf  4711  opelopabf  4712  inopab  4863  cnvopab  5031