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Mirrors > Home > ILE Home > Th. List > syl5reqr | GIF version |
Description: An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.) |
Ref | Expression |
---|---|
syl5reqr.1 | ⊢ 𝐵 = 𝐴 |
syl5reqr.2 | ⊢ (𝜑 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
syl5reqr | ⊢ (𝜑 → 𝐶 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl5reqr.1 | . . 3 ⊢ 𝐵 = 𝐴 | |
2 | 1 | eqcomi 2044 | . 2 ⊢ 𝐴 = 𝐵 |
3 | syl5reqr.2 | . 2 ⊢ (𝜑 → 𝐵 = 𝐶) | |
4 | 2, 3 | syl5req 2085 | 1 ⊢ (𝜑 → 𝐶 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-4 1400 ax-17 1419 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-cleq 2033 |
This theorem is referenced by: bm2.5ii 4222 f1o00 5161 fmpt 5319 fmptsn 5352 resfunexg 5382 prarloclem5 6598 recexprlem1ssl 6731 recexprlem1ssu 6732 iooval2 8784 resqrexlemover 9608 |
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