Theorem inopab 4862

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 3088	. . . 4 ⊢ ([̣z / x]̣([̣w / y]̣φ ∧ [̣w / y]̣ψ) ↔ ([̣z / x]̣[̣w / y]̣φ ∧ [̣z / x]̣[̣w / y]̣ψ))
2		sbcan 3088	. . . . 5 ⊢ ([̣w / y]̣(φ ∧ ψ) ↔ ([̣w / y]̣φ ∧ [̣w / y]̣ψ))
3	2	sbcbii 3101	. . . 4 ⊢ ([̣z / x]̣[̣w / y]̣(φ ∧ ψ) ↔ [̣z / x]̣([̣w / y]̣φ ∧ [̣w / y]̣ψ))
4		opelopabsb 4697	. . . . 5 ⊢ (⟨z, w⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ [̣z / x]̣[̣w / y]̣φ)
5		opelopabsb 4697	. . . . 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 3219	. . 3 ⊢ (⟨z, w⟩ ∈ ({⟨x, y⟩ ∣ φ} ∩ {⟨x, y⟩ ∣ ψ}) ↔ (⟨z, w⟩ ∈ {⟨x, y⟩ ∣ φ} ∧ ⟨z, w⟩ ∈ {⟨x, y⟩ ∣ ψ}))
9		opelopabsb 4697	. . 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 4850	1 ⊢ ({⟨x, y⟩ ∣ φ} ∩ {⟨x, y⟩ ∣ ψ}) = {⟨x, y⟩ ∣ (φ ∧ ψ)}

Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  [̣wsbc 3046   ∩ cin 3208  ⟨cop 4561  {copab 4622

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640

This theorem is referenced by:  inxp  4863  resopab  4999