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Theorem bj-bdfindisg 9336
Description: Version of bj-bdfindis 9335 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 9335 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-bdfindis.bd BOUNDED φ
bj-bdfindis.nf0 xψ
bj-bdfindis.nf1 xχ
bj-bdfindis.nfsuc xθ
bj-bdfindis.0 (x = ∅ → (ψφ))
bj-bdfindis.1 (x = y → (φχ))
bj-bdfindis.suc (x = suc y → (θφ))
bj-bdfindisg.nfa xA
bj-bdfindisg.nfterm xτ
bj-bdfindisg.term (x = A → (φτ))
Assertion
Ref Expression
bj-bdfindisg ((ψ y 𝜔 (χθ)) → (A 𝜔 → τ))
Distinct variable groups:   x,y   φ,y
Allowed substitution hints:   φ(x)   ψ(x,y)   χ(x,y)   θ(x,y)   τ(x,y)   A(x,y)

Proof of Theorem bj-bdfindisg
StepHypRef Expression
1 bj-bdfindis.bd . . 3 BOUNDED φ
2 bj-bdfindis.nf0 . . 3 xψ
3 bj-bdfindis.nf1 . . 3 xχ
4 bj-bdfindis.nfsuc . . 3 xθ
5 bj-bdfindis.0 . . 3 (x = ∅ → (ψφ))
6 bj-bdfindis.1 . . 3 (x = y → (φχ))
7 bj-bdfindis.suc . . 3 (x = suc y → (θφ))
81, 2, 3, 4, 5, 6, 7bj-bdfindis 9335 . 2 ((ψ y 𝜔 (χθ)) → x 𝜔 φ)
9 bj-bdfindisg.nfa . . 3 xA
10 nfcv 2175 . . 3 x𝜔
11 bj-bdfindisg.nfterm . . 3 xτ
12 bj-bdfindisg.term . . 3 (x = A → (φτ))
139, 10, 11, 12bj-rspg 9195 . 2 (x 𝜔 φ → (A 𝜔 → τ))
148, 13syl 14 1 ((ψ y 𝜔 (χθ)) → (A 𝜔 → τ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wnf 1346   wcel 1390  wnfc 2162  wral 2300  c0 3218  suc csuc 4068  𝜔com 4256  BOUNDED wbd 9201
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874  ax-pr 3935  ax-un 4136  ax-bd0 9202  ax-bdor 9205  ax-bdex 9208  ax-bdeq 9209  ax-bdel 9210  ax-bdsb 9211  ax-bdsep 9273  ax-infvn 9329
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257  df-bdc 9230  df-bj-ind 9316
This theorem is referenced by:  bj-nntrans  9339  bj-nnelirr  9341  bj-omtrans  9344
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