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Theorem ceqsalgALT 3529
Description: Alternate proof of ceqsalg 3528, not using ceqsalt 3526. (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 3504 . . 3 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 nfa1 2148 . . . 4 𝑥𝑥(𝑥 = 𝐴𝜑)
3 ceqsalg.1 . . . 4 𝑥𝜓
4 ceqsalg.2 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜓))
54biimpd 231 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
65a2i 14 . . . . 5 ((𝑥 = 𝐴𝜑) → (𝑥 = 𝐴𝜓))
76sps 2176 . . . 4 (∀𝑥(𝑥 = 𝐴𝜑) → (𝑥 = 𝐴𝜓))
82, 3, 7exlimd 2210 . . 3 (∀𝑥(𝑥 = 𝐴𝜑) → (∃𝑥 𝑥 = 𝐴𝜓))
91, 8syl5com 31 . 2 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) → 𝜓))
104biimprcd 252 . . 3 (𝜓 → (𝑥 = 𝐴𝜑))
113, 10alrimi 2205 . 2 (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑))
129, 11impbid1 227 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wal 1528   = wceq 1530  wex 1773  wnf 1777  wcel 2107
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-8 2109  ax-9 2117  ax-10 2138  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1774  df-nf 1778  df-cleq 2812  df-clel 2891
This theorem is referenced by: (None)
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