Theorem sbidd 42459

Description: An identity theorem for substitution. See sbid 2114. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)

Hypothesis

Ref	Expression
sbidd.1	⊢ (𝜑 → [𝑥 / 𝑥]𝜓)

Assertion

Ref	Expression
sbidd	⊢ (𝜑 → 𝜓)

Proof of Theorem sbidd

Step	Hyp	Ref	Expression
1		sbidd.1	. 2 ⊢ (𝜑 → [𝑥 / 𝑥]𝜓)
2		sbid 2114	. 2 ⊢ ([𝑥 / 𝑥]𝜓 ↔ 𝜓)
3	1, 2	sylib 208	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4  [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-sb 1881

This theorem is referenced by: (None)