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Theorem bj-dfssb2 32977
Description: An alternate definition of df-ssb 32958. Note that the use of a dummy variable in the definition df-ssb 32958 allows to use bj-sb56 32976 instead of equs45f 2496 and hence to avoid dependency on ax-13 2408 and to use ax-12 2203 only through bj-ax12 32972. Compare dfsb7 2603. (Contributed by BJ, 25-Dec-2020.)
Assertion
Ref Expression
bj-dfssb2 ([𝑡/𝑥]b𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
Distinct variable groups:   𝑦,𝑡   𝑥,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem bj-dfssb2
StepHypRef Expression
1 df-ssb 32958 . 2 ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
2 bj-sb56 32976 . 2 (∃𝑦(𝑦 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
3 bj-sb56 32976 . . . . 5 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
43bicomi 214 . . . 4 (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝑦𝜑))
54anbi2i 609 . . 3 ((𝑦 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑦𝜑)) ↔ (𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
65exbii 1924 . 2 (∃𝑦(𝑦 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
71, 2, 63bitr2i 288 1 ([𝑡/𝑥]b𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1629  wex 1852  [wssb 32957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-ssb 32958
This theorem is referenced by:  bj-ssbn  32978
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