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Theorem nfimad 4846
Description: Deduction version of bound-variable hypothesis builder nfima 4845. (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 2259 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐴}
2 nfaba1 2259 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐵}
31, 2nfima 4845 . 2 𝑥({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵})
4 nfimad.2 . . 3 (𝜑𝑥𝐴)
5 nfimad.3 . . 3 (𝜑𝑥𝐵)
6 nfnfc1 2256 . . . . 5 𝑥𝑥𝐴
7 nfnfc1 2256 . . . . 5 𝑥𝑥𝐵
86, 7nfan 1525 . . . 4 𝑥(𝑥𝐴𝑥𝐵)
9 abidnf 2819 . . . . . 6 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
109imaeq1d 4836 . . . . 5 (𝑥𝐴 → ({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) = (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧𝐵}))
11 abidnf 2819 . . . . . 6 (𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧𝐵} = 𝐵)
1211imaeq2d 4837 . . . . 5 (𝑥𝐵 → (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) = (𝐴𝐵))
1310, 12sylan9eq 2165 . . . 4 ((𝑥𝐴𝑥𝐵) → ({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) = (𝐴𝐵))
148, 13nfceqdf 2252 . . 3 ((𝑥𝐴𝑥𝐵) → (𝑥({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) ↔ 𝑥(𝐴𝐵)))
154, 5, 14syl2anc 406 . 2 (𝜑 → (𝑥({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) ↔ 𝑥(𝐴𝐵)))
163, 15mpbii 147 1 (𝜑𝑥(𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1310  wcel 1461  {cab 2099  wnfc 2240  cima 4500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-rab 2397  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-xp 4503  df-cnv 4505  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510
This theorem is referenced by: (None)
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