Theorem bnj157 34994

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 ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜑))
bnj157.2	⊢ 𝐴 ∈ V
bnj157.3	⊢ 𝑅 Fr 𝐴

Assertion

Ref	Expression
bnj157	⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) → ∀𝑥 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑅,𝑦   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)

Proof of Theorem bnj157

Step	Hyp	Ref	Expression
1		bnj157.3	. 2 ⊢ 𝑅 Fr 𝐴
2		bnj157.2	. . 3 ⊢ 𝐴 ∈ V
3		bnj157.1	. . 3 ⊢ (𝜓 ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜑))
4	2, 3	bnj110 34993	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → ∀𝑥 ∈ 𝐴 𝜑)
5	1, 4	mpan 691	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2114  ∀wral 3050  Vcvv 3439  [wsbc 3739   class class class wbr 5097   Fr wfr 5573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-fr 5576

This theorem is referenced by:  bnj852  35056