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Theorem ceqsalgALT 3217
Description: Alternate proof of ceqsalg 3216, not using ceqsalt 3214. (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 3201 . . 3 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 nfa1 2025 . . . 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 2053 . . . 4 (∀𝑥(𝑥 = 𝐴𝜑) → (𝑥 = 𝐴𝜓))
82, 3, 7exlimd 2085 . . 3 (∀𝑥(𝑥 = 𝐴𝜑) → (∃𝑥 𝑥 = 𝐴𝜓))
91, 8syl5com 31 . 2 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) → 𝜓))
104biimprcd 240 . . 3 (𝜓 → (𝑥 = 𝐴𝜑))
113, 10alrimi 2080 . 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 1478   = wceq 1480  wex 1701  wnf 1705  wcel 1987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-12 2044  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3188
This theorem is referenced by: (None)
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