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Mirrors > Home > ILE Home > Th. List > ax-gen | GIF version |
Description: Rule of Generalization. The postulated inference rule of predicate calculus. See e.g. Rule 2 of [Hamilton] p. 74. This rule says that if something is unconditionally true, then it is true for all values of a variable. For example, if we have proved 𝑥 = 𝑥, we can conclude ∀𝑥𝑥 = 𝑥 or even ∀𝑦𝑥 = 𝑥. Theorem spi 1516 shows we can go the other way also: in other words we can add or remove universal quantifiers from the beginning of any theorem as required. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
ax-g.1 | ⊢ 𝜑 |
Ref | Expression |
---|---|
ax-gen | ⊢ ∀𝑥𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . 2 wff 𝜑 | |
2 | vx | . 2 setvar 𝑥 | |
3 | 1, 2 | wal 1329 | 1 wff ∀𝑥𝜑 |
Colors of variables: wff set class |
This axiom is referenced by: gen2 1426 mpg 1427 mpgbi 1428 mpgbir 1429 hbth 1439 19.23h 1474 19.9ht 1620 stdpc6 1679 equveli 1732 cesare 2103 camestres 2104 calemes 2115 ceqsralv 2717 vtocl2 2741 euxfr2dc 2869 sbcth 2922 sbciegf 2940 csbiegf 3043 sbcnestg 3053 csbnestg 3054 csbnest1g 3055 int0 3785 mpteq2ia 4014 mpteq2da 4017 ssopab2i 4199 relssi 4630 xpidtr 4929 funcnvsn 5168 funinsn 5172 tfrlem7 6214 tfri1 6262 sucinc 6341 findcard 6782 findcard2 6783 findcard2s 6784 fiintim 6817 fisseneq 6820 frec2uzrand 10178 frec2uzf1od 10179 frecfzennn 10199 hashinfom 10524 zfz1iso 10584 fclim 11063 distop 12254 ch2var 12974 strcollnf 13183 |
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