Theorem opelopabaf 4711

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4709 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Hypotheses

Ref	Expression
opelopabaf.x	⊢ Ⅎxψ
opelopabaf.y	⊢ Ⅎyψ
opelopabaf.1	⊢ A ∈ V
opelopabaf.2	⊢ B ∈ V
opelopabaf.3	⊢ ((x = A ∧ y = B) → (φ ↔ ψ))

Assertion

Ref	Expression
opelopabaf	⊢ (⟨A, B⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ ψ)

Distinct variable groups:   x,y,A   x,B,y

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

Proof of Theorem opelopabaf

Step	Hyp	Ref	Expression
1		opelopabsb 4698	. 2 ⊢ (⟨A, B⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ [̣A / x]̣[̣B / y]̣φ)
2		opelopabaf.1	. . 3 ⊢ A ∈ V
3		opelopabaf.2	. . 3 ⊢ B ∈ V
4		opelopabaf.x	. . . 4 ⊢ Ⅎxψ
5		opelopabaf.y	. . . 4 ⊢ Ⅎyψ
6		nfv 1619	. . . 4 ⊢ Ⅎx B ∈ V
7		opelopabaf.3	. . . 4 ⊢ ((x = A ∧ y = B) → (φ ↔ ψ))
8	4, 5, 6, 7	sbc2iegf 3113	. . 3 ⊢ ((A ∈ V ∧ B ∈ V) → ([̣A / x]̣[̣B / y]̣φ ↔ ψ))
9	2, 3, 8	mp2an 653	. 2 ⊢ ([̣A / x]̣[̣B / y]̣φ ↔ ψ)
10	1, 9	bitri 240	1 ⊢ (⟨A, B⟩ ∈ {⟨x, y⟩ ∣ φ} ↔ ψ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Vcvv 2860  [̣wsbc 3047  ⟨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: (None)