Theorem pm13.195 41920

Description: Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3739. (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 3739	. 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 395   = wceq 1539  ∃wex 1783  [wsbc 3711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-sbc 3712

This theorem is referenced by: (None)