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Theorem aevdemo 26709
 Description: Proof illustrating the comment of aev2 1973. (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 1970 . . . 4 (∀𝑥 𝑥 = 𝑦 → ∀𝑒 𝑓 = 𝑔)
2119.2d 1880 . . 3 (∀𝑥 𝑥 = 𝑦 → ∃𝑒 𝑓 = 𝑔)
32olcd 407 . 2 (∀𝑥 𝑥 = 𝑦 → (∃𝑎𝑏 𝑐 = 𝑑 ∨ ∃𝑒 𝑓 = 𝑔))
4 aev 1970 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑚 𝑚 = 𝑛)
5 aeveq 1969 . . . . 5 (∀𝑚 𝑚 = 𝑛𝑘 = 𝑙)
65a1d 25 . . . 4 (∀𝑚 𝑚 = 𝑛 → (𝑖 = 𝑗𝑘 = 𝑙))
76alrimiv 1842 . . 3 (∀𝑚 𝑚 = 𝑛 → ∀(𝑖 = 𝑗𝑘 = 𝑙))
84, 7syl 17 . 2 (∀𝑥 𝑥 = 𝑦 → ∀(𝑖 = 𝑗𝑘 = 𝑙))
93, 8jca 553 1 (∀𝑥 𝑥 = 𝑦 → ((∃𝑎𝑏 𝑐 = 𝑑 ∨ ∃𝑒 𝑓 = 𝑔) ∧ ∀(𝑖 = 𝑗𝑘 = 𝑙)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696 This theorem is referenced by: (None)
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