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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-equsalvwd | Structured version Visualization version GIF version |
Description: Variant of equsalvw 2013. (Contributed by BJ, 7-Oct-2024.) |
Ref | Expression |
---|---|
bj-equsalvwd.nf0 | ⊢ (𝜑 → ∀𝑥𝜑) |
bj-equsalvwd.nf | ⊢ (𝜑 → Ⅎ'𝑥𝜒) |
bj-equsalvwd.is | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
bj-equsalvwd | ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-equsalvwd.nf0 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | bj-equsalvwd.is | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
3 | 2 | pm5.74da 804 | . . 3 ⊢ (𝜑 → ((𝑥 = 𝑦 → 𝜓) ↔ (𝑥 = 𝑦 → 𝜒))) |
4 | 1, 3 | albidh 1874 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜒))) |
5 | bj-equsalvwd.nf | . . 3 ⊢ (𝜑 → Ⅎ'𝑥𝜒) | |
6 | bj-equsvt 34647 | . . 3 ⊢ (Ⅎ'𝑥𝜒 → (∀𝑥(𝑥 = 𝑦 → 𝜒) ↔ 𝜒)) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜒) ↔ 𝜒)) |
8 | 4, 7 | bitrd 282 | 1 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1541 Ⅎ'wnnf 34591 |
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-6 1976 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-bj-nnf 34592 |
This theorem is referenced by: bj-equsexvwd 34649 bj-sbievwd 34650 |
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