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Mirrors > Home > MPE Home > Th. List > equsalvw | Structured version Visualization version GIF version |
Description: Version of equsalv 2265 with a disjoint variable condition, and of equsal 2420 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsexvw 2002. (Contributed by BJ, 31-May-2019.) |
Ref | Expression |
---|---|
equsalvw.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
equsalvw | ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equsalvw.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | pm5.74i 271 | . . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜓)) |
3 | 2 | albii 1816 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓)) |
4 | equsv 2000 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜓) | |
5 | 3, 4 | bitri 275 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 |
This theorem depends on definitions: df-bi 207 df-ex 1777 |
This theorem is referenced by: equsexvw 2002 equvelv 2028 sb6 2083 sbievwOLD 2092 ax13lem2 2379 reu8 3742 el 5448 asymref2 6140 intirr 6141 fun11 6642 fv3 6925 fpwwe2lem11 10679 bj-dvelimdv 36834 bj-dvelimdv1 36835 undmrnresiss 43594 pm13.192 44406 |
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