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Mirrors > Home > MPE Home > Th. List > equsalvw | Structured version Visualization version GIF version |
Description: Version of equsalv 2257 with a disjoint variable condition, and of equsal 2415 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsexvw 2007. (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 270 | . . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜓)) |
3 | 2 | albii 1820 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓)) |
4 | equsv 2005 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜓) | |
5 | 3, 4 | bitri 274 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 |
This theorem depends on definitions: df-bi 206 df-ex 1781 |
This theorem is referenced by: equsexvw 2007 equvelv 2033 sb6 2087 sbievw 2094 ax13lem2 2374 reu8 3730 el 5438 asymref2 6119 intirr 6120 fun11 6623 fv3 6910 fpwwe2lem11 10639 bj-dvelimdv 36034 bj-dvelimdv1 36035 undmrnresiss 42658 pm13.192 43472 |
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