Theorem setinds2regs 35644

Description: Principle of set induction (or E-induction). If a property passes from all elements of 𝑥 to 𝑥 itself, then it holds for all 𝑥. (Contributed by BTernaryTau, 31-Dec-2025.)

Hypotheses

Ref	Expression
setinds2regs.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
setinds2regs.2	⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)

Assertion

Ref	Expression
setinds2regs	⊢ 𝜑

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦

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

Proof of Theorem setinds2regs

Step	Hyp	Ref	Expression
1		vex 3457	. 2 ⊢ 𝑥 ∈ V
2		setinds2regs.1	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	cbvabv 2832	. . . 4 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
4		setindregs 35643	. . . . 5 ⊢ (∀𝑥(𝑥 ⊆ {𝑦 ∣ 𝜓} → 𝑥 ∈ {𝑦 ∣ 𝜓}) → {𝑦 ∣ 𝜓} = V)
5		ssabral 4015	. . . . . . 7 ⊢ (𝑥 ⊆ {𝑦 ∣ 𝜓} ↔ ∀𝑦 ∈ 𝑥 𝜓)
6		setinds2regs.2	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)
7	5, 6	sylbi 220	. . . . . 6 ⊢ (𝑥 ⊆ {𝑦 ∣ 𝜓} → 𝜑)
8	3	eqabcri 2905	. . . . . 6 ⊢ (𝜑 ↔ 𝑥 ∈ {𝑦 ∣ 𝜓})
9	7, 8	sylib 221	. . . . 5 ⊢ (𝑥 ⊆ {𝑦 ∣ 𝜓} → 𝑥 ∈ {𝑦 ∣ 𝜓})
10	4, 9	mpg 1830	. . . 4 ⊢ {𝑦 ∣ 𝜓} = V
11	3, 10	eqtri 2785	. . 3 ⊢ {𝑥 ∣ 𝜑} = V
12	11	eqabcri 2905	. 2 ⊢ (𝜑 ↔ 𝑥 ∈ V)
13	1, 12	mpbir 234	1 ⊢ 𝜑

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  {cab 2740  ∀wral 3078  Vcvv 3453   ⊆ wss 3902

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-regs 35639

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-v 3455  df-dif 3905  df-in 3909  df-ss 3919  df-nul 4283

This theorem is used by:  tz9.1regs  35647  trssfir1omregs  35649  r1omhfbregs  35650