Theorem sbtd 42191

Description: A true statement is true upon substitution (deduction). A similar proof is possible for icht 47385. (Contributed by SN, 4-May-2024.)

Hypothesis

Ref	Expression
sbtd.1	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
sbtd	⊢ (𝜑 → [𝑡 / 𝑥]𝜓)

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem sbtd

Step	Hyp	Ref	Expression
1		sbtd.1	. . 3 ⊢ (𝜑 → 𝜓)
2	1	alrimiv 1926	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
3		stdpc4 2067	. 2 ⊢ (∀𝑥𝜓 → [𝑡 / 𝑥]𝜓)
4	2, 3	syl 17	1 ⊢ (𝜑 → [𝑡 / 𝑥]𝜓)

Syntax hints:   → wi 4  ∀wal 1537  [wsb 2063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909

This theorem depends on definitions:  df-bi 207  df-sb 2064

This theorem is referenced by: (None)