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Theorem ceqsalgALT 3363
Description: Alternate proof of ceqsalg 3362, not using ceqsalt 3360. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Revised by BJ, 29-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ceqsalg.1 𝑥𝜓
ceqsalg.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsalgALT (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem ceqsalgALT
StepHypRef Expression
1 elisset 3347 . . 3 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 nfa1 2169 . . . 4 𝑥𝑥(𝑥 = 𝐴𝜑)
3 ceqsalg.1 . . . 4 𝑥𝜓
4 ceqsalg.2 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜓))
54biimpd 219 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
65a2i 14 . . . . 5 ((𝑥 = 𝐴𝜑) → (𝑥 = 𝐴𝜓))
76sps 2194 . . . 4 (∀𝑥(𝑥 = 𝐴𝜑) → (𝑥 = 𝐴𝜓))
82, 3, 7exlimd 2226 . . 3 (∀𝑥(𝑥 = 𝐴𝜑) → (∃𝑥 𝑥 = 𝐴𝜓))
91, 8syl5com 31 . 2 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) → 𝜓))
104biimprcd 240 . . 3 (𝜓 → (𝑥 = 𝐴𝜑))
113, 10alrimi 2221 . 2 (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑))
129, 11impbid1 215 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1622   = wceq 1624  wex 1845  wnf 1849  wcel 2131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-12 2188  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-v 3334
This theorem is referenced by: (None)
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