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Definition df-ssb 32745
Description: Alternate definition of proper substitution. Note that the occurrences of a given variable are either all bound (𝑥, 𝑦) or all free (𝑡). Also note that the definiens uses only primitive symbols. It is obtained by applying twice Tarski's definition sb6 2457 which is valid for disjoint variables, so we introduce a dummy variable 𝑦 to isolate 𝑥 from 𝑡, as in dfsb7 2483 with respect to sb5 2458.

This double level definition will make several proofs using it appear as doubled. Alternately, one could often first prove as a lemma the same theorem with a DV condition on the substitute and the substituted variables, and then prove the original theorem by applying this lemma twice in a row.

This definition uses a dummy variable, but the justification theorem, bj-ssbjust 32743, is provable from Tarski's FOL.

Once this is proved, more of the fundamental properties of proper substitution will be provable from Tarski's FOL system, sometimes with the help of specialization sp 2091, of the substitution axiom ax-12 2087, and of commutation of quantifiers ax-11 2074; that is, ax-13 2282 will often be avoided. (Contributed by BJ, 22-Dec-2020.)

Assertion
Ref Expression
df-ssb ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
Distinct variable groups:   𝑥,𝑦   𝑦,𝑡   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑡)

Detailed syntax breakdown of Definition df-ssb
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 vx . . 3 setvar 𝑥
3 vt . . 3 setvar 𝑡
41, 2, 3wssb 32744 . 2 wff [𝑡/𝑥]b𝜑
5 vy . . . . 5 setvar 𝑦
65, 3weq 1931 . . . 4 wff 𝑦 = 𝑡
72, 5weq 1931 . . . . . 6 wff 𝑥 = 𝑦
87, 1wi 4 . . . . 5 wff (𝑥 = 𝑦𝜑)
98, 2wal 1521 . . . 4 wff 𝑥(𝑥 = 𝑦𝜑)
106, 9wi 4 . . 3 wff (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑))
1110, 5wal 1521 . 2 wff 𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑))
124, 11wb 196 1 wff ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
This definition is referenced by:  bj-ssbim  32746  bj-alsb  32750  bj-sbex  32751  bj-ssbeq  32752  bj-ssb0  32753  bj-ssbequ  32754  bj-ssb1a  32757  bj-ssb1  32758  bj-dfssb2  32765  bj-ssbn  32766  bj-ssbequ2  32768  bj-ssbequ1  32769  bj-ssbid2ALT  32771  bj-ssbid1ALT  32773  bj-ssbssblem  32774
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