Theorem setindf 16353

Description: Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness hypothesis. (Contributed by BJ, 22-Nov-2019.)

Hypothesis

Ref	Expression
setindf.nf	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
setindf	⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem setindf

Step	Hyp	Ref	Expression
1		setindft 16352	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑))
2		setindf.nf	. 2 ⊢ Ⅎ𝑦𝜑
3	1, 2	mpg 1497	1 ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1393  Ⅎwnf 1506  [wsb 1808  ∀wral 2508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-cleq 2222  df-clel 2225  df-ral 2513

This theorem is referenced by: (None)