Theorem setinds2regs 35136

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 3440	. 2 ⊢ 𝑥 ∈ V
2		setinds2regs.1	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	cbvabv 2801	. . . 4 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
4		setindregs 35135	. . . . 5 ⊢ (∀𝑥(𝑥 ⊆ {𝑦 ∣ 𝜓} → 𝑥 ∈ {𝑦 ∣ 𝜓}) → {𝑦 ∣ 𝜓} = V)
5		ssabral 4012	. . . . . . 7 ⊢ (𝑥 ⊆ {𝑦 ∣ 𝜓} ↔ ∀𝑦 ∈ 𝑥 𝜓)
6		setinds2regs.2	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)
7	5, 6	sylbi 217	. . . . . 6 ⊢ (𝑥 ⊆ {𝑦 ∣ 𝜓} → 𝜑)
8	3	eqabcri 2875	. . . . . 6 ⊢ (𝜑 ↔ 𝑥 ∈ {𝑦 ∣ 𝜓})
9	7, 8	sylib 218	. . . . 5 ⊢ (𝑥 ⊆ {𝑦 ∣ 𝜓} → 𝑥 ∈ {𝑦 ∣ 𝜓})
10	4, 9	mpg 1798	. . . 4 ⊢ {𝑦 ∣ 𝜓} = V
11	3, 10	eqtri 2754	. . 3 ⊢ {𝑥 ∣ 𝜑} = V
12	11	eqabcri 2875	. 2 ⊢ (𝜑 ↔ 𝑥 ∈ V)
13	1, 12	mpbir 231	1 ⊢ 𝜑

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047  Vcvv 3436   ⊆ wss 3897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-regs 35131

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-v 3438  df-dif 3900  df-in 3904  df-ss 3914  df-nul 4283

This theorem is referenced by:  tz9.1regs  35137  trssfir1omregs  35139  r1omhfbregs  35140