Theorem sbtd 43008

Description: A true statement is true upon substitution (deduction). A similar proof is possible for icht 48229. (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 1957	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
3		stdpc4 2102	. 2 ⊢ (∀𝑥𝜓 → [𝑡 / 𝑥]𝜓)
4	2, 3	syl 18	1 ⊢ (𝜑 → [𝑡 / 𝑥]𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  [wsb 2096

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-sb 2097

This theorem is used by: (None)