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| Mirrors > Home > MPE Home > Th. List > equsalvw | Structured version Visualization version GIF version | ||
| Description: Version of equsalv 2267 with a disjoint variable condition, and of equsal 2422 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsexvw 2004. (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 1819 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓)) | 
| 4 | equsv 2002 | . 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 1538 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 | 
| This theorem depends on definitions: df-bi 207 df-ex 1780 | 
| This theorem is referenced by: equsexvw 2004 equvelv 2030 sb6 2085 sbievwOLD 2094 ax13lem2 2381 reu8 3739 el 5442 asymref2 6137 intirr 6138 fun11 6640 fv3 6924 fpwwe2lem11 10681 bj-dvelimdv 36852 bj-dvelimdv1 36853 undmrnresiss 43617 pm13.192 44429 | 
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