Theorem equs45f 2462

Description: Two ways of expressing substitution when 𝑦 is not free in 𝜑. The implication "to the left" is equs4 2419 and does not require the nonfreeness hypothesis. Theorem sbalex 2241 replaces the nonfreeness hypothesis with a disjoint variable condition and equs5 2463 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 2375. Use sbalex 2241 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 2194	. . . . 5 ⊢ (𝜑 → ∀𝑦𝜑)
3	2	anim2i 617	. . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 ∧ ∀𝑦𝜑))
4	3	eximi 1834	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑))
5		equs5a 2460	. . 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 2419	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
8	6, 7	impbii 209	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1537  ∃wex 1778  Ⅎwnf 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-10 2140  ax-12 2176  ax-13 2375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1779  df-nf 1783

This theorem is referenced by:  sb5f  2501