Theorem pm13.195 42031

Description: Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3744. (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 3744	. 2 ⊢ ([𝐴 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑))
2	1	bicomi 223	1 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑)

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782  [wsbc 3716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-sbc 3717

This theorem is referenced by: (None)