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Mirrors > Home > MPE Home > Th. List > Mathboxes > dral1-o | Structured version Visualization version GIF version |
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). Version of dral1 2461 using ax-c11 36025. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dral1-o.1 | ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
dral1-o | ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbae-o 36041 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑥∀𝑥 𝑥 = 𝑦) | |
2 | dral1-o.1 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | biimpd 231 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → 𝜓)) |
4 | 1, 3 | alimdh 1818 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑥𝜓)) |
5 | ax-c11 36025 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜓 → ∀𝑦𝜓)) | |
6 | 4, 5 | syld 47 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜓)) |
7 | hbae-o 36041 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦∀𝑥 𝑥 = 𝑦) | |
8 | 2 | biimprd 250 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜓 → 𝜑)) |
9 | 7, 8 | alimdh 1818 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 → ∀𝑦𝜑)) |
10 | ax-c11 36025 | . . . 4 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦𝜑 → ∀𝑥𝜑)) | |
11 | 10 | aecoms-o 36040 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑)) |
12 | 9, 11 | syld 47 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜓 → ∀𝑥𝜑)) |
13 | 6, 12 | impbid 214 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-11 2161 ax-c5 36021 ax-c4 36022 ax-c7 36023 ax-c10 36024 ax-c11 36025 ax-c9 36028 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 |
This theorem is referenced by: ax12fromc15 36043 axc16g-o 36072 ax12indalem 36083 ax12inda2ALT 36084 |
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