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Mirrors > Home > MPE Home > Th. List > eqeq12dOLD | Structured version Visualization version GIF version |
Description: Obsolete version of eqeq12d 2753 as of 23-Oct-2024. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
eqeq12dOLD.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
eqeq12dOLD.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
eqeq12dOLD | ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq12dOLD.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | eqeq12dOLD.2 | . 2 ⊢ (𝜑 → 𝐶 = 𝐷) | |
3 | eqeq12OLD 2756 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) | |
4 | 1, 2, 3 | syl2anc 587 | 1 ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-cleq 2729 |
This theorem is referenced by: eqeqan12dOLD 2758 |
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