Theorem indstrd

Description: Strong induction, deduction version. (Contributed by Steven Nguyen, 13-Jul-2025)

	Ref	Expression
Hypotheses	indstrd.1	⊢ ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) )
	indstrd.2	⊢ ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜃 ) )
	indstrd.3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℕ ∧ ∀ 𝑦 ∈ ℕ ( 𝑦 < 𝑥 → 𝜒 ) ) → 𝜓 )
	indstrd.4	⊢ ( 𝜑 → 𝐴 ∈ ℕ )
Assertion	indstrd	⊢ ( 𝜑 → 𝜃 )

Proof

Step	Hyp	Ref	Expression
1		indstrd.1	⊢ ( 𝑥 = 𝑦 → ( 𝜓 ↔ 𝜒 ) )
2		indstrd.2	⊢ ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜃 ) )
3		indstrd.3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℕ ∧ ∀ 𝑦 ∈ ℕ ( 𝑦 < 𝑥 → 𝜒 ) ) → 𝜓 )
4		indstrd.4	⊢ ( 𝜑 → 𝐴 ∈ ℕ )
5		eleq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ ℕ ↔ 𝐴 ∈ ℕ ) )
6	5 2	imbi12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ∈ ℕ → 𝜓 ) ↔ ( 𝐴 ∈ ℕ → 𝜃 ) ) )
7	6	adantl	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( ( 𝑥 ∈ ℕ → 𝜓 ) ↔ ( 𝐴 ∈ ℕ → 𝜃 ) ) )
8	1	imbi2d	⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 → 𝜓 ) ↔ ( 𝜑 → 𝜒 ) ) )
9		bi2.04	⊢ ( ( 𝑦 < 𝑥 → ( 𝜑 → 𝜒 ) ) ↔ ( 𝜑 → ( 𝑦 < 𝑥 → 𝜒 ) ) )
10	9	ralbii	⊢ ( ∀ 𝑦 ∈ ℕ ( 𝑦 < 𝑥 → ( 𝜑 → 𝜒 ) ) ↔ ∀ 𝑦 ∈ ℕ ( 𝜑 → ( 𝑦 < 𝑥 → 𝜒 ) ) )
11		r19.21v	⊢ ( ∀ 𝑦 ∈ ℕ ( 𝜑 → ( 𝑦 < 𝑥 → 𝜒 ) ) ↔ ( 𝜑 → ∀ 𝑦 ∈ ℕ ( 𝑦 < 𝑥 → 𝜒 ) ) )
12	10 11	bitri	⊢ ( ∀ 𝑦 ∈ ℕ ( 𝑦 < 𝑥 → ( 𝜑 → 𝜒 ) ) ↔ ( 𝜑 → ∀ 𝑦 ∈ ℕ ( 𝑦 < 𝑥 → 𝜒 ) ) )
13	3	3com12	⊢ ( ( 𝑥 ∈ ℕ ∧ 𝜑 ∧ ∀ 𝑦 ∈ ℕ ( 𝑦 < 𝑥 → 𝜒 ) ) → 𝜓 )
14	13	3exp	⊢ ( 𝑥 ∈ ℕ → ( 𝜑 → ( ∀ 𝑦 ∈ ℕ ( 𝑦 < 𝑥 → 𝜒 ) → 𝜓 ) ) )
15	14	a2d	⊢ ( 𝑥 ∈ ℕ → ( ( 𝜑 → ∀ 𝑦 ∈ ℕ ( 𝑦 < 𝑥 → 𝜒 ) ) → ( 𝜑 → 𝜓 ) ) )
16	12 15	biimtrid	⊢ ( 𝑥 ∈ ℕ → ( ∀ 𝑦 ∈ ℕ ( 𝑦 < 𝑥 → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜓 ) ) )
17	8 16	indstr	⊢ ( 𝑥 ∈ ℕ → ( 𝜑 → 𝜓 ) )
18	17	com12	⊢ ( 𝜑 → ( 𝑥 ∈ ℕ → 𝜓 ) )
19	4 7 18	vtocld	⊢ ( 𝜑 → ( 𝐴 ∈ ℕ → 𝜃 ) )
20	4 19	mpd	⊢ ( 𝜑 → 𝜃 )