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 Description: Deduction version of bound-variable hypothesis builder nfima 4953. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
Assertion
Ref Expression

Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 nfaba1 2494 . . 3 x{z x z A}
2 nfaba1 2494 . . 3 x{z x z B}
31, 2nfima 4953 . 2 x({z x z A} “ {z x z B})
4 nfimad.2 . . 3 (φxA)
5 nfimad.3 . . 3 (φxB)
6 nfnfc1 2492 . . . . 5 xxA
7 nfnfc1 2492 . . . . 5 xxB
86, 7nfan 1824 . . . 4 x(xA xB)
9 abidnf 3005 . . . . . 6 (xA → {z x z A} = A)
109imaeq1d 4941 . . . . 5 (xA → ({z x z A} “ {z x z B}) = (A “ {z x z B}))
11 abidnf 3005 . . . . . 6 (xB → {z x z B} = B)
1211imaeq2d 4942 . . . . 5 (xB → (A “ {z x z B}) = (AB))
1310, 12sylan9eq 2405 . . . 4 ((xA xB) → ({z x z A} “ {z x z B}) = (AB))
148, 13nfceqdf 2488 . . 3 ((xA xB) → (x({z x z A} “ {z x z B}) ↔ x(AB)))
154, 5, 14syl2anc 642 . 2 (φ → (x({z x z A} “ {z x z B}) ↔ x(AB)))
163, 15mpbii 202 1 (φx(AB))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2476   “ cima 4722 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-br 4640  df-ima 4727 This theorem is referenced by: (None)
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