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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-equsal2 | Structured version Visualization version GIF version |
Description: One direction of equsal 2416. (Contributed by BJ, 30-Sep-2018.) |
Ref | Expression |
---|---|
bj-equsal2.1 | ⊢ Ⅎ𝑥𝜑 |
bj-equsal2.2 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
bj-equsal2 | ⊢ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-equsal2.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | bj-equsal1ti 34743 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑) |
3 | bj-equsal2.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
4 | 3 | a2i 14 | . . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜓)) |
5 | 4 | alimi 1819 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜓)) |
6 | 2, 5 | sylbir 238 | 1 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1541 Ⅎwnf 1791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-12 2175 ax-13 2371 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-nf 1792 |
This theorem is referenced by: bj-equsal 34746 |
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