Theorem wfis2f 5635

Description: Well Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 29-Jan-2011.)

Hypotheses

Ref	Expression
wfis2f.1	⊢ 𝑅 We 𝐴
wfis2f.2	⊢ 𝑅 Se 𝐴
wfis2f.3	⊢ Ⅎ𝑦𝜓
wfis2f.4	⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))
wfis2f.5	⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑))

Assertion

Ref	Expression
wfis2f	⊢ (𝑦 ∈ 𝐴 → 𝜑)

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

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

Proof of Theorem wfis2f

Step	Hyp	Ref	Expression
1		wfis2f.1	. . 3 ⊢ 𝑅 We 𝐴
2		wfis2f.2	. . 3 ⊢ 𝑅 Se 𝐴
3		wfis2f.3	. . . 4 ⊢ Ⅎ𝑦𝜓
4		wfis2f.4	. . . 4 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))
5		wfis2f.5	. . . 4 ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑))
6	3, 4, 5	wfis2fg 5634	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)
7	1, 2, 6	mp2an 704	. 2 ⊢ ∀𝑦 ∈ 𝐴 𝜑
8	7	rspec 2915	1 ⊢ (𝑦 ∈ 𝐴 → 𝜑)

Syntax hints:   → wi 4   ↔ wb 195  Ⅎwnf 1699   ∈ wcel 1977  ∀wral 2896   Se wse 4995   We wwe 4996  Predcpred 5596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597

This theorem is referenced by: (None)