Theorem sbidd-misc 46092

Description: An identity theorem for substitution. See sbid 2253. 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 2253	. 2 ⊢ ([𝑥 / 𝑥]𝜓 ↔ 𝜓)
2	1	imbi2i 339	1 ⊢ ((𝜑 → [𝑥 / 𝑥]𝜓) ↔ (𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 209  [wsb 2070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-sb 2071

This theorem is referenced by: (None)