Theorem sbidd-misc 43372

Description: An identity theorem for substitution. See sbid 2289. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)

Assertion

Ref	Expression
sbidd-misc	⊢ ((𝜑 → [𝑥 / 𝑥]𝜓) ↔ (𝜑 → 𝜓))

Proof of Theorem sbidd-misc

Step	Hyp	Ref	Expression
1		sbid 2289	. 2 ⊢ ([𝑥 / 𝑥]𝜓 ↔ 𝜓)
2	1	imbi2i 328	1 ⊢ ((𝜑 → [𝑥 / 𝑥]𝜓) ↔ (𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 198  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-12 2220

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879  df-sb 2068

This theorem is referenced by: (None)