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| Mirrors > Home > MPE Home > Th. List > eqeqan12rd | Structured version Visualization version GIF version | ||
| Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) |
| Ref | Expression |
|---|---|
| eqeqan12rd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| eqeqan12rd.2 | ⊢ (𝜓 → 𝐶 = 𝐷) |
| Ref | Expression |
|---|---|
| eqeqan12rd | ⊢ ((𝜓 ∧ 𝜑) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqeqan12rd.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | eqeqan12rd.2 | . . 3 ⊢ (𝜓 → 𝐶 = 𝐷) | |
| 3 | 1, 2 | eqeqan12d 2751 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
| 4 | 3 | ancoms 458 | 1 ⊢ ((𝜓 ∧ 𝜑) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 |
| This theorem is referenced by: fmptco 7149 axcontlem4 28982 usgredg4 29234 cusgrsize 29472 uspgr2wlkeqi 29666 clwwlkf1 30068 eigorthi 31856 goeleq12bg 35354 expdiophlem2 43034 pwssplit4 43101 fcoresf1 47081 prproropf1olem4 47493 fmtnoodd 47520 isgrim 47868 |
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