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Theorem nfimad 4705
Description: Deduction version of bound-variable hypothesis builder nfima 4704. (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 2199 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐴}
2 nfaba1 2199 . . 3 𝑥{𝑧 ∣ ∀𝑥 𝑧𝐵}
31, 2nfima 4704 . 2 𝑥({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵})
4 nfimad.2 . . 3 (𝜑𝑥𝐴)
5 nfimad.3 . . 3 (𝜑𝑥𝐵)
6 nfnfc1 2197 . . . . 5 𝑥𝑥𝐴
7 nfnfc1 2197 . . . . 5 𝑥𝑥𝐵
86, 7nfan 1473 . . . 4 𝑥(𝑥𝐴𝑥𝐵)
9 abidnf 2732 . . . . . 6 (𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧𝐴} = 𝐴)
109imaeq1d 4695 . . . . 5 (𝑥𝐴 → ({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) = (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧𝐵}))
11 abidnf 2732 . . . . . 6 (𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧𝐵} = 𝐵)
1211imaeq2d 4696 . . . . 5 (𝑥𝐵 → (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) = (𝐴𝐵))
1310, 12sylan9eq 2108 . . . 4 ((𝑥𝐴𝑥𝐵) → ({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) = (𝐴𝐵))
148, 13nfceqdf 2193 . . 3 ((𝑥𝐴𝑥𝐵) → (𝑥({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) ↔ 𝑥(𝐴𝐵)))
154, 5, 14syl2anc 397 . 2 (𝜑 → (𝑥({𝑧 ∣ ∀𝑥 𝑧𝐴} “ {𝑧 ∣ ∀𝑥 𝑧𝐵}) ↔ 𝑥(𝐴𝐵)))
163, 15mpbii 140 1 (𝜑𝑥(𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257  wcel 1409  {cab 2042  wnfc 2181  cima 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rab 2332  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386
This theorem is referenced by: (None)
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