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Theorem setindtrs 37058
Description: Epsilon induction scheme without Infinity. See comments at setindtr 37057. (Contributed by Stefan O'Rear, 28-Oct-2014.)
Hypotheses
Ref Expression
setindtrs.a (∀𝑦𝑥 𝜓𝜑)
setindtrs.b (𝑥 = 𝑦 → (𝜑𝜓))
setindtrs.c (𝑥 = 𝐵 → (𝜑𝜒))
Assertion
Ref Expression
setindtrs (∃𝑧(Tr 𝑧𝐵𝑧) → 𝜒)
Distinct variable groups:   𝑥,𝐵,𝑧   𝜑,𝑦   𝜓,𝑥   𝜒,𝑥   𝜑,𝑧   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦,𝑧)   𝜒(𝑦,𝑧)   𝐵(𝑦)

Proof of Theorem setindtrs
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 setindtr 37057 . . 3 (∀𝑎(𝑎 ⊆ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑}) → (∃𝑧(Tr 𝑧𝐵𝑧) → 𝐵 ∈ {𝑥𝜑}))
2 dfss3 3578 . . . 4 (𝑎 ⊆ {𝑥𝜑} ↔ ∀𝑦𝑎 𝑦 ∈ {𝑥𝜑})
3 nfcv 2767 . . . . . . 7 𝑥𝑎
4 nfsab1 2616 . . . . . . 7 𝑥 𝑦 ∈ {𝑥𝜑}
53, 4nfral 2945 . . . . . 6 𝑥𝑦𝑎 𝑦 ∈ {𝑥𝜑}
6 nfsab1 2616 . . . . . 6 𝑥 𝑎 ∈ {𝑥𝜑}
75, 6nfim 1827 . . . . 5 𝑥(∀𝑦𝑎 𝑦 ∈ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑})
8 raleq 3132 . . . . . 6 (𝑥 = 𝑎 → (∀𝑦𝑥 𝑦 ∈ {𝑥𝜑} ↔ ∀𝑦𝑎 𝑦 ∈ {𝑥𝜑}))
9 eleq1 2692 . . . . . 6 (𝑥 = 𝑎 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑎 ∈ {𝑥𝜑}))
108, 9imbi12d 334 . . . . 5 (𝑥 = 𝑎 → ((∀𝑦𝑥 𝑦 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜑}) ↔ (∀𝑦𝑎 𝑦 ∈ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑})))
11 setindtrs.a . . . . . 6 (∀𝑦𝑥 𝜓𝜑)
12 vex 3194 . . . . . . . 8 𝑦 ∈ V
13 setindtrs.b . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
1412, 13elab 3338 . . . . . . 7 (𝑦 ∈ {𝑥𝜑} ↔ 𝜓)
1514ralbii 2979 . . . . . 6 (∀𝑦𝑥 𝑦 ∈ {𝑥𝜑} ↔ ∀𝑦𝑥 𝜓)
16 abid 2614 . . . . . 6 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
1711, 15, 163imtr4i 281 . . . . 5 (∀𝑦𝑥 𝑦 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜑})
187, 10, 17chvar 2266 . . . 4 (∀𝑦𝑎 𝑦 ∈ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑})
192, 18sylbi 207 . . 3 (𝑎 ⊆ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑})
201, 19mpg 1721 . 2 (∃𝑧(Tr 𝑧𝐵𝑧) → 𝐵 ∈ {𝑥𝜑})
21 elex 3203 . . . . 5 (𝐵𝑧𝐵 ∈ V)
2221adantl 482 . . . 4 ((Tr 𝑧𝐵𝑧) → 𝐵 ∈ V)
2322exlimiv 1860 . . 3 (∃𝑧(Tr 𝑧𝐵𝑧) → 𝐵 ∈ V)
24 setindtrs.c . . . 4 (𝑥 = 𝐵 → (𝜑𝜒))
2524elabg 3339 . . 3 (𝐵 ∈ V → (𝐵 ∈ {𝑥𝜑} ↔ 𝜒))
2623, 25syl 17 . 2 (∃𝑧(Tr 𝑧𝐵𝑧) → (𝐵 ∈ {𝑥𝜑} ↔ 𝜒))
2720, 26mpbid 222 1 (∃𝑧(Tr 𝑧𝐵𝑧) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1992  {cab 2612  wral 2912  Vcvv 3191  wss 3560  Tr wtr 4717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-reg 8442
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-v 3193  df-dif 3563  df-in 3567  df-ss 3574  df-nul 3897  df-uni 4408  df-tr 4718
This theorem is referenced by:  dford3lem2  37060
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