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Theorem sb6 1782
Description: Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.)
Assertion
Ref Expression
sb6 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb6
StepHypRef Expression
1 sb56 1781 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
21anbi2i 438 . 2 (((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ ((𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝑥 = 𝑦𝜑)))
3 df-sb 1662 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
4 ax-4 1416 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
54pm4.71ri 378 . 2 (∀𝑥(𝑥 = 𝑦𝜑) ↔ ((𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝑥 = 𝑦𝜑)))
62, 3, 53bitr4i 205 1 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257  wex 1397  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-sb 1662
This theorem is referenced by:  sb5  1783  sbnv  1784  sbanv  1785  sbi1v  1787  sbi2v  1788  hbs1  1830  2sb6  1876  sbcom2v  1877  sb6a  1880  sb7af  1885  sbalyz  1891  sbal1yz  1893  exsb  1900  sbal2  1914
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