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Theorem bnj157 34399
Description: Well-founded induction restricted to a set (𝐴 ∈ V). The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj157.1 (𝜓 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))
bnj157.2 𝐴 ∈ V
bnj157.3 𝑅 Fr 𝐴
Assertion
Ref Expression
bnj157 (∀𝑥𝐴 (𝜓𝜑) → ∀𝑥𝐴 𝜑)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑅,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)

Proof of Theorem bnj157
StepHypRef Expression
1 bnj157.3 . 2 𝑅 Fr 𝐴
2 bnj157.2 . . 3 𝐴 ∈ V
3 bnj157.1 . . 3 (𝜓 ↔ ∀𝑦𝐴 (𝑦𝑅𝑥[𝑦 / 𝑥]𝜑))
42, 3bnj110 34398 . 2 ((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴 (𝜓𝜑)) → ∀𝑥𝐴 𝜑)
51, 4mpan 687 1 (∀𝑥𝐴 (𝜓𝜑) → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2098  wral 3055  Vcvv 3468  [wsbc 3772   class class class wbr 5141   Fr wfr 5621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-fr 5624
This theorem is referenced by:  bnj852  34461
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