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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-equsal1t | Structured version Visualization version GIF version |
Description: Duplication of wl-equsal1t 36000, with shorter proof. If one imposes a disjoint variable condition on x,y , then one can use alequexv 2004 and reduce axiom dependencies, and similarly for the following theorems. Note: wl-equsalcom 36001 is also interesting. (Contributed by BJ, 6-Oct-2018.) |
Ref | Expression |
---|---|
bj-equsal1t | ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-alequex 35249 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑) | |
2 | 19.9t 2197 | . . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) | |
3 | 1, 2 | imbitrid 243 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑)) |
4 | nf5r 2187 | . . 3 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) | |
5 | ala1 1815 | . . 3 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
6 | 4, 5 | syl6 35 | . 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
7 | 3, 6 | impbid 211 | 1 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1539 ∃wex 1781 Ⅎwnf 1785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-12 2171 ax-13 2370 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-nf 1786 |
This theorem is referenced by: bj-equsal1ti 35288 |
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