Theorem pm13.195 44432

Description: Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3816. (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 3816	. 2 ⊢ ([𝐴 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝜑))
2	1	bicomi 224	1 ⊢ (∃𝑦(𝑦 = 𝐴 ∧ 𝜑) ↔ [𝐴 / 𝑦]𝜑)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779  [wsbc 3788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-sbc 3789

This theorem is referenced by: (None)