Theorem spsbim 2108

Description: Distribute substitution over implication. Closed form of sbimi 2110. Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) Revise df-sb 2099. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Steven Nguyen, 24-Jul-2023.)

Assertion

Ref	Expression
spsbim	⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))

Proof of Theorem spsbim

Step	Hyp	Ref	Expression
1		stdpc4 2104	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → [𝑡 / 𝑥](𝜑 → 𝜓))
2		sbi1 2107	. 2 ⊢ ([𝑡 / 𝑥](𝜑 → 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))
3	1, 2	syl 18	1 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  [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:  spsbbi  2109  sbimdv  2114  sbimd  2280  mo3  2589  ss2abim  3985  bj-hbsb3t  37792  wl-mo3t  38601  pm11.59  45468  sbiota1  45511