Theorem sbi1 2110

Description: Distribute substitution over implication. (Contributed by NM, 14-May-1993.) Remove dependencies on axioms. (Revised by Steven Nguyen, 24-Jul-2023.) Definition df-sb 2098 changed. (Revised by Wolf Lammen, 5-Jun-2026.)

Assertion

Ref	Expression
sbi1	⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Proof of Theorem sbi1

Dummy variables 𝑢 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbi1lem 2109	. . 3 ⊢ (([𝑦 / 𝑥](𝜑 → 𝜓) ∧ [𝑦 / 𝑥]𝜑) → ∀𝑢(𝑢 = 𝑦 → ∀𝑥(𝑥 = 𝑢 → 𝜓)))
2		sbi1lem 2109	. . 3 ⊢ (([𝑦 / 𝑥](𝜑 → 𝜓) ∧ [𝑦 / 𝑥]𝜑) → ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜓)))
3		df-sb 2098	. . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ (∀𝑢(𝑢 = 𝑦 → ∀𝑥(𝑥 = 𝑢 → 𝜓)) ∧ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜓))))
4	1, 2, 3	sylanbrc 594	. 2 ⊢ (([𝑦 / 𝑥](𝜑 → 𝜓) ∧ [𝑦 / 𝑥]𝜑) → [𝑦 / 𝑥]𝜓)
5	4	ex 417	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1565  [wsb 2097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836

This theorem depends on definitions:  df-bi 210  df-an 401  df-sb 2098

This theorem is referenced by:  spsbim  2112  sbimi  2114  sb2imi  2115  sbrimvw  2131  sbim  2344  sbcim1  3804  2sb5ndVD  45545  2sb5ndALT  45567