Theorem equs45f 2452

Description: Two ways of expressing substitution when 𝑦 is not free in 𝜑. The implication "to the left" is equs4 2409 and does not require the nonfreeness hypothesis. Theorem sbalex 2227 replaces the nonfreeness hypothesis with a disjoint variable condition and equs5 2453 replaces it with a distinctor antecedent. (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 4-Oct-2016.) Usage of this theorem is discouraged because it depends on ax-13 2365. Use sbalex 2227 instead. (New usage is discouraged.)

Hypothesis

Ref	Expression
equs45f.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
equs45f	⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Proof of Theorem equs45f

Step	Hyp	Ref	Expression
1		equs45f.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
2	1	nf5ri 2180	. . . . 5 ⊢ (𝜑 → ∀𝑦𝜑)
3	2	anim2i 616	. . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 ∧ ∀𝑦𝜑))
4	3	eximi 1829	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑))
5		equs5a 2450	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
6	4, 5	syl 17	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
7		equs4 2409	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
8	6, 7	impbii 208	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1531  ∃wex 1773  Ⅎwnf 1777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2163  ax-13 2365

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778

This theorem is referenced by:  sb5f  2491