Theorem spsbim 2077

Description: Distribute substitution over implication. Closed form of sbimi 2079. Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) Revise df-sb 2070. (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 2073	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → [𝑡 / 𝑥](𝜑 → 𝜓))
2		sbi1 2076	. 2 ⊢ ([𝑡 / 𝑥](𝜑 → 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))
3	1, 2	syl 17	1 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))

Syntax hints:   → wi 4  ∀wal 1535  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911

This theorem depends on definitions:  df-bi 209  df-sb 2070

This theorem is referenced by:  spsbbi  2078  sbimdv  2083  sbimd  2245  mo3  2648  bj-hbsb3t  34112  wl-mo3t  34814  pm11.59  40730  sbiota1  40773