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Theorem setindtrs 37909
Description: Epsilon induction scheme without Infinity. See comments at setindtr 37908. (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 37908 . . 3 (∀𝑎(𝑎 ⊆ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑}) → (∃𝑧(Tr 𝑧𝐵𝑧) → 𝐵 ∈ {𝑥𝜑}))
2 dfss3 3625 . . . 4 (𝑎 ⊆ {𝑥𝜑} ↔ ∀𝑦𝑎 𝑦 ∈ {𝑥𝜑})
3 nfcv 2793 . . . . . . 7 𝑥𝑎
4 nfsab1 2641 . . . . . . 7 𝑥 𝑦 ∈ {𝑥𝜑}
53, 4nfral 2974 . . . . . 6 𝑥𝑦𝑎 𝑦 ∈ {𝑥𝜑}
6 nfsab1 2641 . . . . . 6 𝑥 𝑎 ∈ {𝑥𝜑}
75, 6nfim 1865 . . . . 5 𝑥(∀𝑦𝑎 𝑦 ∈ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑})
8 raleq 3168 . . . . . 6 (𝑥 = 𝑎 → (∀𝑦𝑥 𝑦 ∈ {𝑥𝜑} ↔ ∀𝑦𝑎 𝑦 ∈ {𝑥𝜑}))
9 eleq1 2718 . . . . . 6 (𝑥 = 𝑎 → (𝑥 ∈ {𝑥𝜑} ↔ 𝑎 ∈ {𝑥𝜑}))
108, 9imbi12d 333 . . . . 5 (𝑥 = 𝑎 → ((∀𝑦𝑥 𝑦 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜑}) ↔ (∀𝑦𝑎 𝑦 ∈ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑})))
11 setindtrs.a . . . . . 6 (∀𝑦𝑥 𝜓𝜑)
12 vex 3234 . . . . . . . 8 𝑦 ∈ V
13 setindtrs.b . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
1412, 13elab 3382 . . . . . . 7 (𝑦 ∈ {𝑥𝜑} ↔ 𝜓)
1514ralbii 3009 . . . . . 6 (∀𝑦𝑥 𝑦 ∈ {𝑥𝜑} ↔ ∀𝑦𝑥 𝜓)
16 abid 2639 . . . . . 6 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
1711, 15, 163imtr4i 281 . . . . 5 (∀𝑦𝑥 𝑦 ∈ {𝑥𝜑} → 𝑥 ∈ {𝑥𝜑})
187, 10, 17chvar 2298 . . . 4 (∀𝑦𝑎 𝑦 ∈ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑})
192, 18sylbi 207 . . 3 (𝑎 ⊆ {𝑥𝜑} → 𝑎 ∈ {𝑥𝜑})
201, 19mpg 1764 . 2 (∃𝑧(Tr 𝑧𝐵𝑧) → 𝐵 ∈ {𝑥𝜑})
21 elex 3243 . . . . 5 (𝐵𝑧𝐵 ∈ V)
2221adantl 481 . . . 4 ((Tr 𝑧𝐵𝑧) → 𝐵 ∈ V)
2322exlimiv 1898 . . 3 (∃𝑧(Tr 𝑧𝐵𝑧) → 𝐵 ∈ V)
24 setindtrs.c . . . 4 (𝑥 = 𝐵 → (𝜑𝜒))
2524elabg 3383 . . 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 383   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wral 2941  Vcvv 3231  wss 3607  Tr wtr 4785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-reg 8538
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-uni 4469  df-tr 4786
This theorem is referenced by:  dford3lem2  37911
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