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Theorem equs5eALT 2380
Description: Alternate proof of equs5e 2478. Uses ax-12 2169 but not ax-13 2385. (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 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Proof of Theorem equs5eALT
StepHypRef Expression
1 nfa1 2148 . 2 𝑥𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)
2 hbe1 2140 . . . . 5 (∃𝑦𝜑 → ∀𝑦𝑦𝜑)
3219.23bi 2182 . . . 4 (𝜑 → ∀𝑦𝑦𝜑)
4 ax-12 2169 . . . 4 (𝑥 = 𝑦 → (∀𝑦𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)))
53, 4syl5 34 . . 3 (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)))
65imp 407 . 2 ((𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
71, 6exlimi 2210 1 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1528  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-12 2169
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-ex 1774  df-nf 1778
This theorem is referenced by: (None)
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