Theorem equs5e 2481

Description: A property related to substitution that unlike equs5 2483 does not require a distinctor antecedent. This proof uses ax12 2445, see equs5eALT 2385 for an alternative one using ax-12 2177 but not ax13 2393. Usage of this theorem is discouraged because it depends on ax-13 2390. (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 2155	. 2 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)
2		ax12 2445	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑦∃𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)))
3		hbe1 2147	. . . 4 ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑)
4	3	19.23bi 2190	. . 3 ⊢ (𝜑 → ∀𝑦∃𝑦𝜑)
5	2, 4	impel 508	. 2 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
6	1, 5	exlimi 2217	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1535  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785

This theorem is referenced by:  sb4e  2524