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Mirrors > Home > MPE Home > Th. List > eqeq12OLD | Structured version Visualization version GIF version |
Description: Obsolete version of eqeq12 2743 as of 23-Oct-2024. (Contributed by NM, 3-Aug-1994.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
eqeq12OLD | ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2730 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶)) | |
2 | eqeq2 2738 | . 2 ⊢ (𝐶 = 𝐷 → (𝐵 = 𝐶 ↔ 𝐵 = 𝐷)) | |
3 | 1, 2 | sylan9bb 509 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-cleq 2718 |
This theorem is referenced by: eqeq12dOLD 2746 |
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