Theorem sb56 2276

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 2275 and sb6 2092. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 2069. The implication "to the left" is equs4 2437 and does not require any disjoint variable condition (but the version with a disjoint variable condition, equs4v 2005, requires fewer axioms). Theorem equs45f 2481 replaces the disjoint variable condition with a non-freeness hypothesis and equs5 2482 replaces it with a distinctor as antecedent. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2268 in place of equsex 2439 in order to remove dependency on ax-13 2389. (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 2275	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
2		sb6 2092	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	1, 2	bitr3i 279	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1534  ∃wex 1779  [wsb 2068

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 1969  ax-7 2014  ax-10 2144  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1780  df-nf 1784  df-sb 2069

This theorem is referenced by:  sb5OLD  2280  dfsb7  2284  dfsb7OLD  2285  sb4vOLDOLD  2512  sb4vOLDALT  2583  sb5ALT2  2585  mopick  2709  alexeqg  3647  dfdif3  4094  pm13.196a  40752