Theorem sb56OLD 2279

Description: Obsolete version of sbalex 2243 as of 21-Sep-2024. Two equivalent ways of expressing the proper substitution of 𝑦 for 𝑥 in 𝜑, when 𝑥 and 𝑦 are distinct, namely, alternate definitions sb5 2277 and sb6 2085. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 2065. The implication "to the left" is equs4 2424 and does not require any disjoint variable condition (but the version with a disjoint variable condition, equs4v 1999, requires fewer axioms). Theorem equs45f 2467 replaces the disjoint variable condition with a nonfreeness hypothesis and equs5 2468 replaces it with a distinctor as antecedent. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2269 in place of equsex 2426 in order to remove dependency on ax-13 2380. (Revised by BJ, 20-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sb56OLD	⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb56OLD

Step	Hyp	Ref	Expression
1		sb5 2277	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
2		sb6 2085	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	1, 2	bitr3i 277	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535  ∃wex 1777  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-nf 1782  df-sb 2065

This theorem is referenced by: (None)