Theorem aevdemo 27317

Description: Proof illustrating the comment of aev2 1986. (Contributed by BJ, 30-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
aevdemo	⊢ (∀𝑥 𝑥 = 𝑦 → ((∃𝑎∀𝑏 𝑐 = 𝑑 ∨ ∃𝑒 𝑓 = 𝑔) ∧ ∀ℎ(𝑖 = 𝑗 → 𝑘 = 𝑙)))

Distinct variable group:   𝑥,𝑦

Proof of Theorem aevdemo

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aev 1983	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑒 𝑓 = 𝑔)
2	1	19.2d 1893	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑒 𝑓 = 𝑔)
3	2	olcd 408	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑎∀𝑏 𝑐 = 𝑑 ∨ ∃𝑒 𝑓 = 𝑔))
4		aev 1983	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑚 𝑚 = 𝑛)
5		aeveq 1982	. . . . 5 ⊢ (∀𝑚 𝑚 = 𝑛 → 𝑘 = 𝑙)
6	5	a1d 25	. . . 4 ⊢ (∀𝑚 𝑚 = 𝑛 → (𝑖 = 𝑗 → 𝑘 = 𝑙))
7	6	alrimiv 1855	. . 3 ⊢ (∀𝑚 𝑚 = 𝑛 → ∀ℎ(𝑖 = 𝑗 → 𝑘 = 𝑙))
8	4, 7	syl 17	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀ℎ(𝑖 = 𝑗 → 𝑘 = 𝑙))
9	3, 8	jca 554	1 ⊢ (∀𝑥 𝑥 = 𝑦 → ((∃𝑎∀𝑏 𝑐 = 𝑑 ∨ ∃𝑒 𝑓 = 𝑔) ∧ ∀ℎ(𝑖 = 𝑗 → 𝑘 = 𝑙)))

Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705

This theorem is referenced by: (None)