Theorem sb2imi 2111

Description: Distribute substitution over implication. Compare al2imi 1847. (Contributed by Steven Nguyen, 13-Aug-2023.)

Hypothesis

Ref	Expression
sb2imi.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
sb2imi	⊢ ([𝑡 / 𝑥]𝜑 → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜒))

Proof of Theorem sb2imi

Step	Hyp	Ref	Expression
1		sb2imi.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	sbimi 2110	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥](𝜓 → 𝜒))
3		sbi1 2107	. 2 ⊢ ([𝑡 / 𝑥](𝜓 → 𝜒) → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜒))
4	2, 3	syl 18	1 ⊢ ([𝑡 / 𝑥]𝜑 → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜒))

This proof depends on syntax axioms:   → wi 4  [wsb 2098

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099

This theorem is used by:  sban  2116  sbn1  2143