Theorem equs5eALT 2365

Description: Alternate proof of equs5e 2458. Uses ax-12 2172 but not ax-13 2372. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
equs5eALT	⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Proof of Theorem equs5eALT

Step	Hyp	Ref	Expression
1		nfa1 2149	. 2 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)
2		hbe1 2140	. . . . 5 ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑)
3	2	19.23bi 2185	. . . 4 ⊢ (𝜑 → ∀𝑦∃𝑦𝜑)
4		ax-12 2172	. . . 4 ⊢ (𝑥 = 𝑦 → (∀𝑦∃𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)))
5	3, 4	syl5 34	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)))
6	5	imp 408	. 2 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
7	1, 6	exlimi 2211	1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Syntax hints:   → wi 4   ∧ wa 397  ∀wal 1540  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-12 2172

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787

This theorem is referenced by: (None)