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Theorem aevdemo 27205
 Description: Proof illustrating the comment of aev2 1983. (Contributed by BJ, 30-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
aevdemo (∀𝑥 𝑥 = 𝑦 → ((∃𝑎𝑏 𝑐 = 𝑑 ∨ ∃𝑒 𝑓 = 𝑔) ∧ ∀(𝑖 = 𝑗𝑘 = 𝑙)))
Distinct variable group:   𝑥,𝑦

Proof of Theorem aevdemo
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 aev 1980 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∀𝑒 𝑓 = 𝑔)
2119.2d 1890 . . 3 (∀𝑥 𝑥 = 𝑦 → ∃𝑒 𝑓 = 𝑔)
32olcd 408 . 2 (∀𝑥 𝑥 = 𝑦 → (∃𝑎𝑏 𝑐 = 𝑑 ∨ ∃𝑒 𝑓 = 𝑔))
4 aev 1980 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑚 𝑚 = 𝑛)
5 aeveq 1979 . . . . 5 (∀𝑚 𝑚 = 𝑛𝑘 = 𝑙)
65a1d 25 . . . 4 (∀𝑚 𝑚 = 𝑛 → (𝑖 = 𝑗𝑘 = 𝑙))
76alrimiv 1852 . . 3 (∀𝑚 𝑚 = 𝑛 → ∀(𝑖 = 𝑗𝑘 = 𝑙))
84, 7syl 17 . 2 (∀𝑥 𝑥 = 𝑦 → ∀(𝑖 = 𝑗𝑘 = 𝑙))
93, 8jca 554 1 (∀𝑥 𝑥 = 𝑦 → ((∃𝑎𝑏 𝑐 = 𝑑 ∨ ∃𝑒 𝑓 = 𝑔) ∧ ∀(𝑖 = 𝑗𝑘 = 𝑙)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384  ∀wal 1478  ∃wex 1701 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 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702 This theorem is referenced by: (None)
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