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Theorem sb6 1854
 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 1853 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
21anbi2i 453 . 2 (((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ ((𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝑥 = 𝑦𝜑)))
3 df-sb 1732 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
4 ax-4 1487 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
54pm4.71ri 390 . 2 (∀𝑥(𝑥 = 𝑦𝜑) ↔ ((𝑥 = 𝑦𝜑) ∧ ∀𝑥(𝑥 = 𝑦𝜑)))
62, 3, 53bitr4i 211 1 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1330  ∃wex 1469  [wsb 1731 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-11 1483  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511 This theorem depends on definitions:  df-bi 116  df-sb 1732 This theorem is referenced by:  sb5  1855  sbnv  1856  sbanv  1857  sbi1v  1859  sbi2v  1860  hbs1  1906  2sb6  1952  sbcom2v  1953  sb6a  1956  sb7af  1961  sbalyz  1967  sbal1yz  1969  exsb  1976  sbal2  1988
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