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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-equsalv | Structured version Visualization version GIF version |
Description: Version of equsal 2279 with a dv condition, which does not require ax-13 2234. See equsalvw 1918 for a version with two dv conditions requiring fewer axioms. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-equsalv.nf | ⊢ Ⅎ𝑥𝜓 |
bj-equsalv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
bj-equsalv | ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-equsalv.nf | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | 1 | 19.23 2067 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ (∃𝑥 𝑥 = 𝑦 → 𝜓)) |
3 | bj-equsalv.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 3 | pm5.74i 259 | . . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜓)) |
5 | 4 | albii 1737 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓)) |
6 | ax6ev 1877 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
7 | 6 | a1bi 351 | . 2 ⊢ (𝜓 ↔ (∃𝑥 𝑥 = 𝑦 → 𝜓)) |
8 | 2, 5, 7 | 3bitr4i 291 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 ∃wex 1695 Ⅎwnf 1699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-or 384 df-ex 1696 df-nf 1701 |
This theorem is referenced by: bj-equsalhv 31932 |
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