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Mirrors > Home > ILE Home > Th. List > eq2tri | GIF version |
Description: A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.) |
Ref | Expression |
---|---|
eq2tr.1 | ⊢ (𝐴 = 𝐶 → 𝐷 = 𝐹) |
eq2tr.2 | ⊢ (𝐵 = 𝐷 → 𝐶 = 𝐺) |
Ref | Expression |
---|---|
eq2tri | ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐹) ↔ (𝐵 = 𝐷 ∧ 𝐴 = 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 266 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (𝐵 = 𝐷 ∧ 𝐴 = 𝐶)) | |
2 | eq2tr.1 | . . . 4 ⊢ (𝐴 = 𝐶 → 𝐷 = 𝐹) | |
3 | 2 | eqeq2d 2189 | . . 3 ⊢ (𝐴 = 𝐶 → (𝐵 = 𝐷 ↔ 𝐵 = 𝐹)) |
4 | 3 | pm5.32i 454 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐹)) |
5 | eq2tr.2 | . . . 4 ⊢ (𝐵 = 𝐷 → 𝐶 = 𝐺) | |
6 | 5 | eqeq2d 2189 | . . 3 ⊢ (𝐵 = 𝐷 → (𝐴 = 𝐶 ↔ 𝐴 = 𝐺)) |
7 | 6 | pm5.32i 454 | . 2 ⊢ ((𝐵 = 𝐷 ∧ 𝐴 = 𝐶) ↔ (𝐵 = 𝐷 ∧ 𝐴 = 𝐺)) |
8 | 1, 4, 7 | 3bitr3i 210 | 1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐹) ↔ (𝐵 = 𝐷 ∧ 𝐴 = 𝐺)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 ax-4 1510 ax-17 1526 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-cleq 2170 |
This theorem is referenced by: xpassen 6833 |
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