Theorem equs5e 2438

Description: A property related to substitution that unlike equs5 2440 does not require a distinctor antecedent. See equs5eALT 2342 for an alternate proof using ax-12 2141 but not ax13 2347. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.)

Assertion

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

Proof of Theorem equs5e

Step	Hyp	Ref	Expression
1		nfa1 2121	. 2 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)
2		ax12 2402	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑦∃𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)))
3		hbe1 2114	. . . 4 ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑)
4	3	19.23bi 2154	. . 3 ⊢ (𝜑 → ∀𝑦∃𝑦𝜑)
5	2, 4	impel 506	. 2 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
6	1, 5	exlimi 2182	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1520  ∃wex 1761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-12 2141  ax-13 2344

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1762  df-nf 1766

This theorem is referenced by:  sb4e  2478