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Mirrors > Home > ILE Home > Th. List > stdpc5 | GIF version |
Description: An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis Ⅎ𝑥𝜑 can be thought of as emulating "𝑥 is not free in 𝜑." With this definition, the meaning of "not free" is less restrictive than the usual textbook definition; for example 𝑥 would not (for us) be free in 𝑥 = 𝑥 by nfequid 1630. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) |
Ref | Expression |
---|---|
stdpc5.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
stdpc5 | ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stdpc5.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | 19.21 1515 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) |
3 | 2 | biimpi 118 | 1 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1282 Ⅎwnf 1389 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-4 1440 ax-ial 1467 ax-i5r 1468 |
This theorem depends on definitions: df-bi 115 df-nf 1390 |
This theorem is referenced by: sbalyz 1916 ra5 2902 |
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