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Theorem sb56 1857
Description: Two equivalent ways of expressing the proper substitution of 𝑦 for 𝑥 in 𝜑, when 𝑥 and 𝑦 are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1736. (Contributed by NM, 14-Apr-2008.)
Assertion
Ref Expression
sb56 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb56
StepHypRef Expression
1 hba1 1520 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥𝑥(𝑥 = 𝑦𝜑))
2 ax11v 1799 . . 3 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
3 ax-4 1487 . . . 4 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
43com12 30 . . 3 (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦𝜑) → 𝜑))
52, 4impbid 128 . 2 (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
61, 5equsex 1706 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1329  wex 1468
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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  sb6  1858  sb5  1859  alexeq  2806  dfdif3  3181
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