Theorem sbimdv 2082

Description: Deduction substituting both sides of an implication, with 𝜑 and 𝑥 disjoint. See also sbimd 2240. (Contributed by Wolf Lammen, 6-May-2023.) Revise df-sb 2069. (Revised by Steven Nguyen, 6-Jul-2023.)

Hypothesis

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

Assertion

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

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem sbimdv

Step	Hyp	Ref	Expression
1		sbimdv.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	alrimiv 1931	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
3		spsbim 2076	. 2 ⊢ (∀𝑥(𝜓 → 𝜒) → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜒))
4	2, 3	syl 17	1 ⊢ (𝜑 → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜒))

Syntax hints:   → wi 4  ∀wal 1537  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914

This theorem depends on definitions:  df-bi 206  df-sb 2069

This theorem is referenced by:  sbcimdv  3786  ss2abdv  3993  ss2abdvALT  3994