Theorem equs5av 2275

Description: A property related to substitution that replaces the distinctor from equs5 2459 to a disjoint variable condition. Version of equs5a 2456 with a disjoint variable condition, which does not require ax-13 2371. See also sbalex 2240. (Contributed by NM, 2-Feb-2007.) (Revised by Gino Giotto, 15-Dec-2023.)

Assertion

Ref	Expression
equs5av	⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem equs5av

Step	Hyp	Ref	Expression
1		nfa1 2152	. 2 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑦 → 𝜑)
2		ax12v2 2177	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3	2	spsd 2184	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
4	3	imp 410	. 2 ⊢ ((𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
5	1, 4	exlimi 2215	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1541  ∃wex 1787

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

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:  bj-equs45fv  34730