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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-equsal1 | Structured version Visualization version GIF version |
Description: One direction of equsal 2415. (Contributed by BJ, 30-Sep-2018.) |
Ref | Expression |
---|---|
bj-equsal1.1 | ⊢ Ⅎ𝑥𝜓 |
bj-equsal1.2 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
bj-equsal1 | ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-equsal1.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
2 | 1 | a2i 14 | . . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜓)) |
3 | 2 | alimi 1812 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜓)) |
4 | bj-equsal1.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
5 | 4 | bj-equsal1ti 36005 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜓) |
6 | 3, 5 | sylib 217 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1538 Ⅎwnf 1784 |
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 1912 ax-6 1970 ax-7 2010 ax-12 2170 ax-13 2370 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1781 df-nf 1785 |
This theorem is referenced by: bj-equsal 36008 |
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