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Theorem bj-bdfindisg 7170
 Description: Version of bj-bdfindis 7169 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-bdfindis 7169 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 7169 . 2 ((ψ y 𝜔 (χθ)) → x 𝜔 φ)
9 bj-bdfindisg.nfa . . 3 xA
10 nfcv 2160 . . 3 x𝜔
11 bj-bdfindisg.nfterm . . 3 xτ
12 bj-bdfindisg.term . . 3 (x = A → (φτ))
139, 10, 11, 12bj-rspg 7033 . 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 1228  Ⅎwnf 1329   ∈ wcel 1374  Ⅎwnfc 2147  ∀wral 2284  ∅c0 3201  suc csuc 4051  𝜔com 4240  BOUNDED wbd 7039 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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-nul 3857  ax-pr 3918  ax-un 4120  ax-bd0 7040  ax-bdor 7043  ax-bdex 7046  ax-bdeq 7047  ax-bdel 7048  ax-bdsb 7049  ax-bdsep 7111  ax-infvn 7163 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-iom 4241  df-bdc 7068  df-bj-ind 7150 This theorem is referenced by:  bj-nntrans  7173  bj-nnelirr  7175  bj-omtrans  7178
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