ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfimad GIF version

Theorem nfimad 4860
Description: Deduction version of bound-variable hypothesis builder nfima 4859. (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 2264 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐴}
2 nfaba1 2264 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐵}
31, 2nfima 4859 . 2 𝑥({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵})
4 nfimad.2 . . 3 (𝜑𝑥𝐴)
5 nfimad.3 . . 3 (𝜑𝑥𝐵)
6 nfnfc1 2261 . . . . 5 𝑥𝑥𝐴
7 nfnfc1 2261 . . . . 5 𝑥𝑥𝐵
86, 7nfan 1529 . . . 4 𝑥(𝑥𝐴𝑥𝐵)
9 abidnf 2825 . . . . . 6 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
109imaeq1d 4850 . . . . 5 (𝑥𝐴 → ({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) = (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧𝐵}))
11 abidnf 2825 . . . . . 6 (𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧𝐵} = 𝐵)
1211imaeq2d 4851 . . . . 5 (𝑥𝐵 → (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) = (𝐴𝐵))
1310, 12sylan9eq 2170 . . . 4 ((𝑥𝐴𝑥𝐵) → ({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) = (𝐴𝐵))
148, 13nfceqdf 2257 . . 3 ((𝑥𝐴𝑥𝐵) → (𝑥({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) ↔ 𝑥(𝐴𝐵)))
154, 5, 14syl2anc 408 . 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 1314  wcel 1465  {cab 2103  wnfc 2245  cima 4512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rab 2402  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator