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Theorem cbvopab1 4632
 Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
cbvopab1.1 zφ
cbvopab1.2 xψ
cbvopab1.3 (x = z → (φψ))
Assertion
Ref Expression
cbvopab1 {x, y φ} = {z, y ψ}
Distinct variable groups:   x,y   y,z
Allowed substitution hints:   φ(x,y,z)   ψ(x,y,z)

Proof of Theorem cbvopab1
Dummy variables w v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . . . 5 vy(w = x, y φ)
2 nfv 1619 . . . . . . 7 x w = v, y
3 nfs1v 2106 . . . . . . 7 x[v / x]φ
42, 3nfan 1824 . . . . . 6 x(w = v, y [v / x]φ)
54nfex 1843 . . . . 5 xy(w = v, y [v / x]φ)
6 opeq1 4578 . . . . . . . 8 (x = vx, y = v, y)
76eqeq2d 2364 . . . . . . 7 (x = v → (w = x, yw = v, y))
8 sbequ12 1919 . . . . . . 7 (x = v → (φ ↔ [v / x]φ))
97, 8anbi12d 691 . . . . . 6 (x = v → ((w = x, y φ) ↔ (w = v, y [v / x]φ)))
109exbidv 1626 . . . . 5 (x = v → (y(w = x, y φ) ↔ y(w = v, y [v / x]φ)))
111, 5, 10cbvex 1985 . . . 4 (xy(w = x, y φ) ↔ vy(w = v, y [v / x]φ))
12 nfv 1619 . . . . . . 7 z w = v, y
13 cbvopab1.1 . . . . . . . 8 zφ
1413nfsb 2109 . . . . . . 7 z[v / x]φ
1512, 14nfan 1824 . . . . . 6 z(w = v, y [v / x]φ)
1615nfex 1843 . . . . 5 zy(w = v, y [v / x]φ)
17 nfv 1619 . . . . 5 vy(w = z, y ψ)
18 opeq1 4578 . . . . . . . 8 (v = zv, y = z, y)
1918eqeq2d 2364 . . . . . . 7 (v = z → (w = v, yw = z, y))
20 sbequ 2060 . . . . . . . 8 (v = z → ([v / x]φ ↔ [z / x]φ))
21 cbvopab1.2 . . . . . . . . 9 xψ
22 cbvopab1.3 . . . . . . . . 9 (x = z → (φψ))
2321, 22sbie 2038 . . . . . . . 8 ([z / x]φψ)
2420, 23syl6bb 252 . . . . . . 7 (v = z → ([v / x]φψ))
2519, 24anbi12d 691 . . . . . 6 (v = z → ((w = v, y [v / x]φ) ↔ (w = z, y ψ)))
2625exbidv 1626 . . . . 5 (v = z → (y(w = v, y [v / x]φ) ↔ y(w = z, y ψ)))
2716, 17, 26cbvex 1985 . . . 4 (vy(w = v, y [v / x]φ) ↔ zy(w = z, y ψ))
2811, 27bitri 240 . . 3 (xy(w = x, y φ) ↔ zy(w = z, y ψ))
2928abbii 2465 . 2 {w xy(w = x, y φ)} = {w zy(w = z, y ψ)}
30 df-opab 4623 . 2 {x, y φ} = {w xy(w = x, y φ)}
31 df-opab 4623 . 2 {z, y ψ} = {w zy(w = z, y ψ)}
3229, 30, 313eqtr4i 2383 1 {x, y φ} = {z, y ψ}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  [wsb 1648  {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:  cbvopab1v  4635  cbvmpt  5676
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