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Mirrors > Home > MPE Home > Th. List > aevdemo | Structured version Visualization version GIF version |
Description: Proof illustrating the comment of aev2 2068. (Contributed by BJ, 30-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
aevdemo | ⊢ (∀𝑥 𝑥 = 𝑦 → ((∃𝑎∀𝑏 𝑐 = 𝑑 ∨ ∃𝑒 𝑓 = 𝑔) ∧ ∀ℎ(𝑖 = 𝑗 → 𝑘 = 𝑙))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aev 2067 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑒 𝑓 = 𝑔) | |
2 | 1 | 19.2d 1987 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∃𝑒 𝑓 = 𝑔) |
3 | 2 | olcd 873 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑎∀𝑏 𝑐 = 𝑑 ∨ ∃𝑒 𝑓 = 𝑔)) |
4 | aev 2067 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑚 𝑚 = 𝑛) | |
5 | aeveq 2066 | . . . . 5 ⊢ (∀𝑚 𝑚 = 𝑛 → 𝑘 = 𝑙) | |
6 | 5 | a1d 25 | . . . 4 ⊢ (∀𝑚 𝑚 = 𝑛 → (𝑖 = 𝑗 → 𝑘 = 𝑙)) |
7 | 6 | alrimiv 1934 | . . 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 515 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → ((∃𝑎∀𝑏 𝑐 = 𝑑 ∨ ∃𝑒 𝑓 = 𝑔) ∧ ∀ℎ(𝑖 = 𝑗 → 𝑘 = 𝑙))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 846 ∀wal 1540 ∃wex 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ex 1787 |
This theorem is referenced by: (None) |
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