MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfimad Structured version   Visualization version   GIF version

Theorem nfimad 6026
Description: Deduction version of bound-variable hypothesis builder nfima 6025. (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 2912 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐴}
2 nfaba1 2912 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐵}
31, 2nfima 6025 . 2 𝑥({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵})
4 nfimad.2 . . 3 (𝜑𝑥𝐴)
5 nfimad.3 . . 3 (𝜑𝑥𝐵)
6 nfnfc1 2907 . . . . 5 𝑥𝑥𝐴
7 nfnfc1 2907 . . . . 5 𝑥𝑥𝐵
86, 7nfan 1903 . . . 4 𝑥(𝑥𝐴𝑥𝐵)
9 abidnf 3664 . . . . . 6 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
109imaeq1d 6016 . . . . 5 (𝑥𝐴 → ({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) = (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧𝐵}))
11 abidnf 3664 . . . . . 6 (𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧𝐵} = 𝐵)
1211imaeq2d 6017 . . . . 5 (𝑥𝐵 → (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) = (𝐴𝐵))
1310, 12sylan9eq 2793 . . . 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 585 . 2 (𝜑 → (𝑥({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) ↔ 𝑥(𝐴𝐵)))
163, 15mpbii 232 1 (𝜑𝑥(𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540  wcel 2107  {cab 2710  wnfc 2884  cima 5640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator