Theorem equs45f 2458

Description: Two ways of expressing substitution when 𝑦 is not free in 𝜑. The implication "to the left" is equs4 2415 and does not require the nonfreeness hypothesis. Theorem sbalex 2240 replaces the nonfreeness hypothesis with a disjoint variable condition and equs5 2459 replaces it with a distinctor as antecedent. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 4-Oct-2016.) (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 2193	. . . . 5 ⊢ (𝜑 → ∀𝑦𝜑)
3	2	anim2i 620	. . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 ∧ ∀𝑦𝜑))
4	3	eximi 1842	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑))
5		equs5a 2456	. . 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 2415	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
8	6, 7	impbii 212	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541  ∃wex 1787  Ⅎwnf 1791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2141  ax-12 2175  ax-13 2371

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ex 1788  df-nf 1792

This theorem is referenced by:  sb5f  2501