Theorem setinds2regs 35372

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 3448	. 2 ⊢ 𝑥 ∈ V
2		setinds2regs.1	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	cbvabv 2822	. . . 4 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
4		setindregs 35371	. . . . 5 ⊢ (∀𝑥(𝑥 ⊆ {𝑦 ∣ 𝜓} → 𝑥 ∈ {𝑦 ∣ 𝜓}) → {𝑦 ∣ 𝜓} = V)
5		ssabral 4008	. . . . . . 7 ⊢ (𝑥 ⊆ {𝑦 ∣ 𝜓} ↔ ∀𝑦 ∈ 𝑥 𝜓)
6		setinds2regs.2	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)
7	5, 6	sylbi 219	. . . . . 6 ⊢ (𝑥 ⊆ {𝑦 ∣ 𝜓} → 𝜑)
8	3	eqabcri 2895	. . . . . 6 ⊢ (𝜑 ↔ 𝑥 ∈ {𝑦 ∣ 𝜓})
9	7, 8	sylib 220	. . . . 5 ⊢ (𝑥 ⊆ {𝑦 ∣ 𝜓} → 𝑥 ∈ {𝑦 ∣ 𝜓})
10	4, 9	mpg 1807	. . . 4 ⊢ {𝑦 ∣ 𝜓} = V
11	3, 10	eqtri 2775	. . 3 ⊢ {𝑥 ∣ 𝜑} = V
12	11	eqabcri 2895	. 2 ⊢ (𝜑 ↔ 𝑥 ∈ V)
13	1, 12	mpbir 233	1 ⊢ 𝜑

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1550   ∈ wcel 2132  {cab 2730  ∀wral 3066  Vcvv 3444   ⊆ wss 3895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-regs 35367

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-v 3446  df-dif 3898  df-in 3902  df-ss 3912  df-nul 4277

This theorem is referenced by:  tz9.1regs  35375  trssfir1omregs  35377  r1omhfbregs  35378