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Definition df-ssb 32604
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 2428 which is valid for disjoint variables, so we introduce a dummy variable 𝑦 to isolate 𝑥 from 𝑡, as in dfsb7 2454 with respect to sb5 2429.

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 32602, 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 2052, of the substitution axiom ax-12 2046, and of commutation of quantifiers ax-11 2033; that is, ax-13 2245 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 32603 . 2 wff [𝑡/𝑥]b𝜑
5 vy . . . . 5 setvar 𝑦
65, 3weq 1873 . . . 4 wff 𝑦 = 𝑡
72, 5weq 1873 . . . . . 6 wff 𝑥 = 𝑦
87, 1wi 4 . . . . 5 wff (𝑥 = 𝑦𝜑)
98, 2wal 1480 . . . 4 wff 𝑥(𝑥 = 𝑦𝜑)
106, 9wi 4 . . 3 wff (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑))
1110, 5wal 1480 . 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  32605  bj-alsb  32609  bj-sbex  32610  bj-ssbeq  32611  bj-ssb0  32612  bj-ssbequ  32613  bj-ssb1a  32616  bj-ssb1  32617  bj-dfssb2  32624  bj-ssbn  32625  bj-ssbequ2  32627  bj-ssbequ1  32628  bj-ssbid2ALT  32630  bj-ssbid1ALT  32632  bj-ssbssblem  32633
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