New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  cbvopab2 GIF version

Theorem cbvopab2 4633
 Description: Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)
Hypotheses
Ref Expression
cbvopab2.1 zφ
cbvopab2.2 yψ
cbvopab2.3 (y = z → (φψ))
Assertion
Ref Expression
cbvopab2 {x, y φ} = {x, z ψ}
Distinct variable group:   x,y,z
Allowed substitution hints:   φ(x,y,z)   ψ(x,y,z)

Proof of Theorem cbvopab2
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . . . . 6 z w = x, y
2 cbvopab2.1 . . . . . 6 zφ
31, 2nfan 1824 . . . . 5 z(w = x, y φ)
4 nfv 1619 . . . . . 6 y w = x, z
5 cbvopab2.2 . . . . . 6 yψ
64, 5nfan 1824 . . . . 5 y(w = x, z ψ)
7 opeq2 4579 . . . . . . 7 (y = zx, y = x, z)
87eqeq2d 2364 . . . . . 6 (y = z → (w = x, yw = x, z))
9 cbvopab2.3 . . . . . 6 (y = z → (φψ))
108, 9anbi12d 691 . . . . 5 (y = z → ((w = x, y φ) ↔ (w = x, z ψ)))
113, 6, 10cbvex 1985 . . . 4 (y(w = x, y φ) ↔ z(w = x, z ψ))
1211exbii 1582 . . 3 (xy(w = x, y φ) ↔ xz(w = x, z ψ))
1312abbii 2465 . 2 {w xy(w = x, y φ)} = {w xz(w = x, z ψ)}
14 df-opab 4623 . 2 {x, y φ} = {w xy(w = x, y φ)}
15 df-opab 4623 . 2 {x, z ψ} = {w xz(w = x, z ψ)}
1613, 14, 153eqtr4i 2383 1 {x, y φ} = {x, z ψ}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  {cab 2339  ⟨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-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-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  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-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-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623 This theorem is referenced by:  cbvoprab3  5571
 Copyright terms: Public domain W3C validator