Theorem inopab 4863

Description: Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)

Assertion

Ref	Expression
inopab	⊢ ({⟨x, y⟩ ∣ φ} ∩ {⟨x, y⟩ ∣ ψ}) = {⟨x, y⟩ ∣ (φ ∧ ψ)}

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem inopab

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

Step	Hyp	Ref	Expression
1		sbcan 3089	. . . 4 ⊢ ([̣z / x]̣([̣w / y]̣φ ∧ [̣w / y]̣ψ) ↔ ([̣z / x]̣[̣w / y]̣φ ∧ [̣z / x]̣[̣w / y]̣ψ))
2		sbcan 3089	. . . . 5 ⊢ ([̣w / y]̣(φ ∧ ψ) ↔ ([̣w / y]̣φ ∧ [̣w / y]̣ψ))
3	2	sbcbii 3102	. . . 4 ⊢ ([̣z / x]̣[̣w / y]̣(φ ∧ ψ) ↔ [̣z / x]̣([̣w / y]̣φ ∧ [̣w / y]̣ψ))
4		opelopabsb 4698	. . . . 5 ⊢ (⟨z, w⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ [̣z / x]̣[̣w / y]̣φ)
5		opelopabsb 4698	. . . . 5 ⊢ (⟨z, w⟩ ∈ {⟨x, y⟩ ∣ ψ} ↔ [̣z / x]̣[̣w / y]̣ψ)
6	4, 5	anbi12i 678	. . . 4 ⊢ ((⟨z, w⟩ ∈ {⟨x, y⟩ ∣ φ} ∧ ⟨z, w⟩ ∈ {⟨x, y⟩ ∣ ψ}) ↔ ([̣z / x]̣[̣w / y]̣φ ∧ [̣z / x]̣[̣w / y]̣ψ))
7	1, 3, 6	3bitr4ri 269	. . 3 ⊢ ((⟨z, w⟩ ∈ {⟨x, y⟩ ∣ φ} ∧ ⟨z, w⟩ ∈ {⟨x, y⟩ ∣ ψ}) ↔ [̣z / x]̣[̣w / y]̣(φ ∧ ψ))
8		elin 3220	. . 3 ⊢ (⟨z, w⟩ ∈ ({⟨x, y⟩ ∣ φ} ∩ {⟨x, y⟩ ∣ ψ}) ↔ (⟨z, w⟩ ∈ {⟨x, y⟩ ∣ φ} ∧ ⟨z, w⟩ ∈ {⟨x, y⟩ ∣ ψ}))
9		opelopabsb 4698	. . 3 ⊢ (⟨z, w⟩ ∈ {⟨x, y⟩ ∣ (φ ∧ ψ)} ↔ [̣z / x]̣[̣w / y]̣(φ ∧ ψ))
10	7, 8, 9	3bitr4i 268	. 2 ⊢ (⟨z, w⟩ ∈ ({⟨x, y⟩ ∣ φ} ∩ {⟨x, y⟩ ∣ ψ}) ↔ ⟨z, w⟩ ∈ {⟨x, y⟩ ∣ (φ ∧ ψ)})
11	10	eqrelriv 4851	1 ⊢ ({⟨x, y⟩ ∣ φ} ∩ {⟨x, y⟩ ∣ ψ}) = {⟨x, y⟩ ∣ (φ ∧ ψ)}

Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  [̣wsbc 3047   ∩ cin 3209  ⟨cop 4562  {copab 4623

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:  inxp  4864  resopab  5000