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Theorem bdsetindis 8349
Description: Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bdsetindis.bd BOUNDED φ
bdsetindis.nf0 xψ
bdsetindis.nf1 xχ
bdsetindis.nf2 yφ
bdsetindis.nf3 yψ
bdsetindis.1 (x = z → (φψ))
bdsetindis.2 (x = y → (χφ))
Assertion
Ref Expression
bdsetindis (y(z y ψχ) → xφ)
Distinct variable groups:   x,y,z   φ,z
Allowed substitution hints:   φ(x,y)   ψ(x,y,z)   χ(x,y,z)

Proof of Theorem bdsetindis
StepHypRef Expression
1 nfcv 2151 . . . . 5 xy
2 bdsetindis.nf0 . . . . 5 xψ
31, 2nfralxy 2329 . . . 4 xz y ψ
4 bdsetindis.nf1 . . . 4 xχ
53, 4nfim 1437 . . 3 x(z y ψχ)
6 nfcv 2151 . . . . 5 yx
7 bdsetindis.nf3 . . . . 5 yψ
86, 7nfralxy 2329 . . . 4 yz x ψ
9 bdsetindis.nf2 . . . 4 yφ
108, 9nfim 1437 . . 3 y(z x ψφ)
11 raleq 2474 . . . . 5 (y = x → (z y ψz x ψ))
1211biimprd 147 . . . 4 (y = x → (z x ψz y ψ))
13 bdsetindis.2 . . . . 5 (x = y → (χφ))
1413equcoms 1567 . . . 4 (y = x → (χφ))
1512, 14imim12d 68 . . 3 (y = x → ((z y ψχ) → (z x ψφ)))
165, 10, 15cbv3 1603 . 2 (y(z y ψχ) → x(z x ψφ))
17 bdsetindis.1 . . . . . 6 (x = z → (φψ))
182, 17bj-sbime 8178 . . . . 5 ([z / x]φψ)
1918ralimi 2353 . . . 4 (z x [z / x]φz x ψ)
2019imim1i 54 . . 3 ((z x ψφ) → (z x [z / x]φφ))
2120alimi 1317 . 2 (x(z x ψφ) → x(z x [z / x]φφ))
22 bdsetindis.bd . . 3 BOUNDED φ
2322ax-bdsetind 8348 . 2 (x(z x [z / x]φφ) → xφ)
2416, 21, 233syl 17 1 (y(z y ψχ) → xφ)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1221  wnf 1322  [wsb 1618  wral 2275  BOUNDED wbd 8197
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-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-bdsetind 8348
This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280
This theorem is referenced by:  bj-inf2vnlem3  8352
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