Theorem equs5a 2454

Description: A property related to substitution that unlike equs5 2457 does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2369. This proof uses ax12 2420, see equs5aALT 2361 for an alternative one using ax-12 2169 but not ax13 2372. Usage of the weaker equs5av 2268 is preferred, which uses ax12v2 2171, but not ax-13 2369. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)

Assertion

Ref	Expression
equs5a	⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))

Proof of Theorem equs5a

Step	Hyp	Ref	Expression
1		nfa1 2146	. 2 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑦 → 𝜑)
2		ax12 2420	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3	2	imp 405	. 2 ⊢ ((𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
4	1, 3	exlimi 2208	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 394  ∀wal 1537  ∃wex 1779

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 1911  ax-6 1969  ax-7 2009  ax-10 2135  ax-12 2169  ax-13 2369

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-ex 1780  df-nf 1784

This theorem is referenced by:  equs45f  2456