Theorem eloprabg 5580

Description: The law of concretion for operation class abstraction. Compare elopab 4697. (Contributed by set.mm contributors, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.) Removed unnecessary distinct variable requirements. (Revised by set.mm contributors, 19-Dec-2013.)

Hypotheses

Ref	Expression
eloprabg.1	⊢ (x = A → (φ ↔ ψ))
eloprabg.2	⊢ (y = B → (ψ ↔ χ))
eloprabg.3	⊢ (z = C → (χ ↔ θ))

Assertion

Ref	Expression
eloprabg	⊢ ((A ∈ V ∧ B ∈ W ∧ C ∈ X) → (⟨⟨A, B⟩, C⟩ ∈ {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ θ))

Distinct variable groups:   x,y,z,A   x,B,y,z   x,C,y,z   θ,x,y,z

Allowed substitution hints:   φ(x,y,z)   ψ(x,y,z)   χ(x,y,z)   V(x,y,z)   W(x,y,z)   X(x,y,z)

Proof of Theorem eloprabg

Step	Hyp	Ref	Expression
1		eloprabg.1	. . 3 ⊢ (x = A → (φ ↔ ψ))
2		eloprabg.2	. . 3 ⊢ (y = B → (ψ ↔ χ))
3		eloprabg.3	. . 3 ⊢ (z = C → (χ ↔ θ))
4	1, 2, 3	syl3an9b 1250	. 2 ⊢ ((x = A ∧ y = B ∧ z = C) → (φ ↔ θ))
5	4	eloprabga 5579	1 ⊢ ((A ∈ V ∧ B ∈ W ∧ C ∈ X) → (⟨⟨A, B⟩, C⟩ ∈ {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ θ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ⟨cop 4562  {coprab 5528

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-oprab 5529

This theorem is referenced by:  ov  5596  ovg  5602  oprabid2  5647