Theorem sbi1lem 2092

Description: Lemma for sbi1 2093. The core of the proof was extracted from a proof of SN. (Contributed by Wolf Lammen, 5-Jun-2026.)

Assertion

Ref	Expression
sbi1lem	⊢ (([𝑡 / 𝑥](𝜑 → 𝜓) ∧ [𝑡 / 𝑥]𝜑) → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜓)))

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

Allowed substitution hints:   𝜑(𝑥,𝑡)   𝜓(𝑥,𝑡)

Proof of Theorem sbi1lem

Step	Hyp	Ref	Expression
1		dfsbimp 2082	. . 3 ⊢ ([𝑡 / 𝑥](𝜑 → 𝜓) → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))))
2		dfsbimp 2082	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3		ax-2 7	. . . . . 6 ⊢ ((𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜓)))
4	3	al2imi 1825	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
5	4	imim3i 64	. . . 4 ⊢ ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜓))))
6	5	al2imi 1825	. . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜓))))
7	1, 2, 6	syl2im 40	. 2 ⊢ ([𝑡 / 𝑥](𝜑 → 𝜓) → ([𝑡 / 𝑥]𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜓))))
8	7	imp 409	1 ⊢ (([𝑡 / 𝑥](𝜑 → 𝜓) ∧ [𝑡 / 𝑥]𝜑) → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜓)))

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1548  [wsb 2080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819

This theorem depends on definitions:  df-bi 209  df-an 399  df-sb 2081

This theorem is referenced by:  sbi1  2093