Theorem equs5e 2487

Description: A property related to substitution that unlike equs5 2489 does not require a distinctor antecedent. This proof uses ax12 2452, see equs5eALT 2396 for an alternative one using ax-12 2212 but not ax13 2404. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.) (New usage is discouraged.)

Assertion

Ref	Expression
equs5e	⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Proof of Theorem equs5e

Step	Hyp	Ref	Expression
1		nfa1 2187	. 2 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)
2		ax12 2452	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑦∃𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)))
3		hbe1 2179	. . . 4 ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑)
4	3	19.23bi 2227	. . 3 ⊢ (𝜑 → ∀𝑦∃𝑦𝜑)
5	2, 4	impel 515	. 2 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
6	1, 5	exlimi 2253	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212  ax-13 2401

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816

This theorem is used by:  sb4e  2514