Theorem frege58bid 39035

Description: If ∀𝑥𝜑 is affirmed, 𝜑 cannot be denied. Identical to sp 2224. See ax-frege58b 39034 and frege58c 39054 for versions which more closely track the original. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege58bid	⊢ (∀𝑥𝜑 → 𝜑)

Proof of Theorem frege58bid

Step	Hyp	Ref	Expression
1		ax-frege58b 39034	. 2 ⊢ (∀𝑥𝜑 → [𝑥 / 𝑥]𝜑)
2		sbid 2289	. . 3 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
3	2	biimpi 208	. 2 ⊢ ([𝑥 / 𝑥]𝜑 → 𝜑)
4	1, 3	syl 17	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:   → wi 4  ∀wal 1654  [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  ax-frege58b 39034

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

This theorem is referenced by: (None)