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Theorem nfimad 4620
Description: Deduction version of bound-variable hypothesis builder nfima 4619. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
nfimad.2 (φxA)
nfimad.3 (φxB)
Assertion
Ref Expression
nfimad (φx(AB))

Proof of Theorem nfimad
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 nfaba1 2180 . . 3 x{zx z A}
2 nfaba1 2180 . . 3 x{zx z B}
31, 2nfima 4619 . 2 x({zx z A} “ {zx z B})
4 nfimad.2 . . 3 (φxA)
5 nfimad.3 . . 3 (φxB)
6 nfnfc1 2178 . . . . 5 xxA
7 nfnfc1 2178 . . . . 5 xxB
86, 7nfan 1454 . . . 4 x(xA xB)
9 abidnf 2703 . . . . . 6 (xA → {zx z A} = A)
109imaeq1d 4610 . . . . 5 (xA → ({zx z A} “ {zx z B}) = (A “ {zx z B}))
11 abidnf 2703 . . . . . 6 (xB → {zx z B} = B)
1211imaeq2d 4611 . . . . 5 (xB → (A “ {zx z B}) = (AB))
1310, 12sylan9eq 2089 . . . 4 ((xA xB) → ({zx z A} “ {zx z B}) = (AB))
148, 13nfceqdf 2174 . . 3 ((xA xB) → (x({zx z A} “ {zx z B}) ↔ x(AB)))
154, 5, 14syl2anc 391 . 2 (φ → (x({zx z A} “ {zx z B}) ↔ x(AB)))
163, 15mpbii 136 1 (φx(AB))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   wcel 1390  {cab 2023  wnfc 2162  cima 4291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by: (None)
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