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Theorem cbvoprab1 5567
 Description: Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.)
Hypotheses
Ref Expression
cbvoprab1.1 wφ
cbvoprab1.2 xψ
cbvoprab1.3 (x = w → (φψ))
Assertion
Ref Expression
cbvoprab1 {x, y, z φ} = {w, y, z ψ}
Distinct variable group:   x,y,z,w
Allowed substitution hints:   φ(x,y,z,w)   ψ(x,y,z,w)

Proof of Theorem cbvoprab1
Dummy variable v is distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . . . . 6 w v = x, y
2 cbvoprab1.1 . . . . . 6 wφ
31, 2nfan 1824 . . . . 5 w(v = x, y φ)
43nfex 1843 . . . 4 wy(v = x, y φ)
5 nfv 1619 . . . . . 6 x v = w, y
6 cbvoprab1.2 . . . . . 6 xψ
75, 6nfan 1824 . . . . 5 x(v = w, y ψ)
87nfex 1843 . . . 4 xy(v = w, y ψ)
9 opeq1 4578 . . . . . . 7 (x = wx, y = w, y)
109eqeq2d 2364 . . . . . 6 (x = w → (v = x, yv = w, y))
11 cbvoprab1.3 . . . . . 6 (x = w → (φψ))
1210, 11anbi12d 691 . . . . 5 (x = w → ((v = x, y φ) ↔ (v = w, y ψ)))
1312exbidv 1626 . . . 4 (x = w → (y(v = x, y φ) ↔ y(v = w, y ψ)))
144, 8, 13cbvex 1985 . . 3 (xy(v = x, y φ) ↔ wy(v = w, y ψ))
1514opabbii 4626 . 2 {v, z xy(v = x, y φ)} = {v, z wy(v = w, y ψ)}
16 dfoprab2 5558 . 2 {x, y, z φ} = {v, z xy(v = x, y φ)}
17 dfoprab2 5558 . 2 {w, y, z ψ} = {v, z wy(v = w, y ψ)}
1815, 16, 173eqtr4i 2383 1 {x, y, z φ} = {w, y, z ψ}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  ⟨cop 4561  {copab 4622  {coprab 5527 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  df-oprab 5528 This theorem is referenced by: (None)
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