Theorem frege58bid 44490

Description: If ∀𝑥𝜑 is affirmed, 𝜑 cannot be denied. Identical to sp 2221. See ax-frege58b 44489 and frege58c 44509 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 44489	. 2 ⊢ (∀𝑥𝜑 → [𝑥 / 𝑥]𝜑)
2		sbid 2293	. . 3 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
3	2	biimpi 219	. 2 ⊢ ([𝑥 / 𝑥]𝜑 → 𝜑)
4	1, 3	syl 18	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:   → wi 4  ∀wal 1561  [wsb 2093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-12 2215  ax-frege58b 44489

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803  df-sb 2094

This theorem is referenced by: (None)