Theorem pm13.195 37636

Description: Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3427. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)

Assertion

Ref	Expression
pm13.195	⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑)

Distinct variable group:   𝑦,𝐴

Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem pm13.195

Step	Hyp	Ref	Expression
1		sbc5 3427	. 2 ⊢ ([𝐴 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑))
2	1	bicomi 213	1 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑)

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695  [wsbc 3402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403

This theorem is referenced by: (None)