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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-equsal | Structured version Visualization version GIF version |
Description: Shorter proof of equsal 2416. (Contributed by BJ, 30-Sep-2018.) Proof modification is discouraged to avoid using equsal 2416, but "min */exc equsal" is ok. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-equsal.1 | ⊢ Ⅎ𝑥𝜓 |
bj-equsal.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
bj-equsal | ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-equsal.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | bj-equsal.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | biimpd 232 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
4 | 1, 3 | bj-equsal1 34693 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜓) |
5 | 2 | biimprd 251 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑)) |
6 | 1, 5 | bj-equsal2 34694 | . 2 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
7 | 4, 6 | impbii 212 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀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 2018 ax-12 2177 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: (None) |
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