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Theorem cbvopab 4630
 Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)
Hypotheses
Ref Expression
cbvopab.1 zφ
cbvopab.2 wφ
cbvopab.3 xψ
cbvopab.4 yψ
cbvopab.5 ((x = z y = w) → (φψ))
Assertion
Ref Expression
cbvopab {x, y φ} = {z, w ψ}
Distinct variable group:   x,y,z,w
Allowed substitution hints:   φ(x,y,z,w)   ψ(x,y,z,w)

Proof of Theorem cbvopab
Dummy variable v is distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . . . 5 z v = x, y
2 cbvopab.1 . . . . 5 zφ
31, 2nfan 1824 . . . 4 z(v = x, y φ)
4 nfv 1619 . . . . 5 w v = x, y
5 cbvopab.2 . . . . 5 wφ
64, 5nfan 1824 . . . 4 w(v = x, y φ)
7 nfv 1619 . . . . 5 x v = z, w
8 cbvopab.3 . . . . 5 xψ
97, 8nfan 1824 . . . 4 x(v = z, w ψ)
10 nfv 1619 . . . . 5 y v = z, w
11 cbvopab.4 . . . . 5 yψ
1210, 11nfan 1824 . . . 4 y(v = z, w ψ)
13 opeq12 4580 . . . . . 6 ((x = z y = w) → x, y = z, w)
1413eqeq2d 2364 . . . . 5 ((x = z y = w) → (v = x, yv = z, w))
15 cbvopab.5 . . . . 5 ((x = z y = w) → (φψ))
1614, 15anbi12d 691 . . . 4 ((x = z y = w) → ((v = x, y φ) ↔ (v = z, w ψ)))
173, 6, 9, 12, 16cbvex2 2005 . . 3 (xy(v = x, y φ) ↔ zw(v = z, w ψ))
1817abbii 2465 . 2 {v xy(v = x, y φ)} = {v zw(v = z, w ψ)}
19 df-opab 4623 . 2 {x, y φ} = {v xy(v = x, y φ)}
20 df-opab 4623 . 2 {z, w ψ} = {v zw(v = z, w ψ)}
2118, 19, 203eqtr4i 2383 1 {x, y φ} = {z, w 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-1 5  ax-2 6  ax-3 7  ax-mp 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:  cbvopabv  4631
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