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Theorem nfimad 6089
Description: Deduction version of bound-variable hypothesis builder nfima 6088. (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 2911 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐴}
2 nfaba1 2911 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐵}
31, 2nfima 6088 . 2 𝑥({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵})
4 nfimad.2 . . 3 (𝜑𝑥𝐴)
5 nfimad.3 . . 3 (𝜑𝑥𝐵)
6 nfnfc1 2906 . . . . 5 𝑥𝑥𝐴
7 nfnfc1 2906 . . . . 5 𝑥𝑥𝐵
86, 7nfan 1897 . . . 4 𝑥(𝑥𝐴𝑥𝐵)
9 abidnf 3711 . . . . . 6 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
109imaeq1d 6079 . . . . 5 (𝑥𝐴 → ({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) = (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧𝐵}))
11 abidnf 3711 . . . . . 6 (𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧𝐵} = 𝐵)
1211imaeq2d 6080 . . . . 5 (𝑥𝐵 → (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) = (𝐴𝐵))
1310, 12sylan9eq 2795 . . . 4 ((𝑥𝐴𝑥𝐵) → ({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) = (𝐴𝐵))
148, 13nfceqdf 2899 . . 3 ((𝑥𝐴𝑥𝐵) → (𝑥({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) ↔ 𝑥(𝐴𝐵)))
154, 5, 14syl2anc 584 . 2 (𝜑 → (𝑥({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) ↔ 𝑥(𝐴𝐵)))
163, 15mpbii 233 1 (𝜑𝑥(𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535  wcel 2106  {cab 2712  wnfc 2888  cima 5692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702
This theorem is referenced by: (None)
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