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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege58bid | Structured version Visualization version GIF version |
Description: If ∀𝑥𝜑 is affirmed, 𝜑 cannot be denied. Identical to sp 2053. See ax-frege58b 38195 and frege58c 38215 for versions which more closely track the original. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege58bid | ⊢ (∀𝑥𝜑 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-frege58b 38195 | . 2 ⊢ (∀𝑥𝜑 → [𝑥 / 𝑥]𝜑) | |
2 | sbid 2114 | . . 3 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) | |
3 | 2 | biimpi 206 | . 2 ⊢ ([𝑥 / 𝑥]𝜑 → 𝜑) |
4 | 1, 3 | syl 17 | 1 ⊢ (∀𝑥𝜑 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1481 [wsb 1880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-12 2047 ax-frege58b 38195 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-sb 1881 |
This theorem is referenced by: (None) |
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