Theorem frege58bid 40241

Description: If ∀𝑥𝜑 is affirmed, 𝜑 cannot be denied. Identical to sp 2178. See ax-frege58b 40240 and frege58c 40260 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 40240	. 2 ⊢ (∀𝑥𝜑 → [𝑥 / 𝑥]𝜑)
2		sbid 2253	. . 3 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
3	2	biimpi 218	. 2 ⊢ ([𝑥 / 𝑥]𝜑 → 𝜑)
4	1, 3	syl 17	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:   → wi 4  ∀wal 1531  [wsb 2065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-12 2173  ax-frege58b 40240

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-sb 2066

This theorem is referenced by: (None)