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Theorem cbvmpt 5676
 Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.)
Hypotheses
Ref Expression
cbvmpt.1 yB
cbvmpt.2 xC
cbvmpt.3 (x = yB = C)
Assertion
Ref Expression
cbvmpt (x A B) = (y A C)
Distinct variable groups:   x,A   y,A
Allowed substitution hints:   B(x,y)   C(x,y)

Proof of Theorem cbvmpt
Dummy variables z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . . 4 w(x A z = B)
2 nfv 1619 . . . . 5 x w A
3 nfs1v 2106 . . . . 5 x[w / x]z = B
42, 3nfan 1824 . . . 4 x(w A [w / x]z = B)
5 eleq1 2413 . . . . 5 (x = w → (x Aw A))
6 sbequ12 1919 . . . . 5 (x = w → (z = B ↔ [w / x]z = B))
75, 6anbi12d 691 . . . 4 (x = w → ((x A z = B) ↔ (w A [w / x]z = B)))
81, 4, 7cbvopab1 4632 . . 3 {x, z (x A z = B)} = {w, z (w A [w / x]z = B)}
9 nfv 1619 . . . . 5 y w A
10 cbvmpt.1 . . . . . . 7 yB
1110nfeq2 2500 . . . . . 6 y z = B
1211nfsb 2109 . . . . 5 y[w / x]z = B
139, 12nfan 1824 . . . 4 y(w A [w / x]z = B)
14 nfv 1619 . . . 4 w(y A z = C)
15 eleq1 2413 . . . . 5 (w = y → (w Ay A))
16 sbequ 2060 . . . . . 6 (w = y → ([w / x]z = B ↔ [y / x]z = B))
17 cbvmpt.2 . . . . . . . 8 xC
1817nfeq2 2500 . . . . . . 7 x z = C
19 cbvmpt.3 . . . . . . . 8 (x = yB = C)
2019eqeq2d 2364 . . . . . . 7 (x = y → (z = Bz = C))
2118, 20sbie 2038 . . . . . 6 ([y / x]z = Bz = C)
2216, 21syl6bb 252 . . . . 5 (w = y → ([w / x]z = Bz = C))
2315, 22anbi12d 691 . . . 4 (w = y → ((w A [w / x]z = B) ↔ (y A z = C)))
2413, 14, 23cbvopab1 4632 . . 3 {w, z (w A [w / x]z = B)} = {y, z (y A z = C)}
258, 24eqtri 2373 . 2 {x, z (x A z = B)} = {y, z (y A z = C)}
26 df-mpt 5652 . 2 (x A B) = {x, z (x A z = B)}
27 df-mpt 5652 . 2 (y A C) = {y, z (y A z = C)}
2825, 26, 273eqtr4i 2383 1 (x A B) = (y A C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642  [wsb 1648   ∈ wcel 1710  Ⅎwnfc 2476  {copab 4622   ↦ cmpt 5651 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-mpt 5652 This theorem is referenced by:  cbvmptv  5677  fvmpts  5701  fvmpt2i  5703  fvmptex  5721
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