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Mirrors > Home > ILE Home > Th. List > syl5req | GIF version |
Description: An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
Ref | Expression |
---|---|
syl5req.1 | ⊢ 𝐴 = 𝐵 |
syl5req.2 | ⊢ (𝜑 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
syl5req | ⊢ (𝜑 → 𝐶 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl5req.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
2 | syl5req.2 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐶) | |
3 | 1, 2 | syl5eq 2184 | . 2 ⊢ (𝜑 → 𝐴 = 𝐶) |
4 | 3 | eqcomd 2145 | 1 ⊢ (𝜑 → 𝐶 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-gen 1425 ax-4 1487 ax-17 1506 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-cleq 2132 |
This theorem is referenced by: syl5reqr 2187 opeqsn 4174 dcextest 4495 relop 4689 funopg 5157 funcnvres 5196 mapsnconst 6588 snexxph 6838 apreap 8349 recextlem1 8412 nn0supp 9029 intqfrac2 10092 hashprg 10554 hashfacen 10579 explecnv 11274 rerestcntop 12719 |
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