Theorem sb56 2274

Description: Two equivalent ways of expressing the proper substitution of 𝑦 for 𝑥 in 𝜑, when 𝑥 and 𝑦 are distinct, namely, alternate definitions sb5 2273 and sb6 2090. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 2070. The implication "to the left" is equs4 2427 and does not require any disjoint variable condition (but the version with a disjoint variable condition, equs4v 2006, requires fewer axioms). Theorem equs45f 2471 replaces the disjoint variable condition with a non-freeness hypothesis and equs5 2472 replaces it with a distinctor as antecedent. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2266 in place of equsex 2429 in order to remove dependency on ax-13 2379. (Revised by BJ, 20-Dec-2020.)

Assertion

Ref	Expression
sb56	⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb56

Step	Hyp	Ref	Expression
1		sb5 2273	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
2		sb6 2090	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	1, 2	bitr3i 280	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070

This theorem is referenced by:  sb5OLD  2277  dfsb7  2281  dfsb7OLD  2282  sb4vOLDALT  2560  sb5ALT2  2562  mopick  2687  alexeqg  3592  dfdif3  4042  pm13.196a  41118