![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2747 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
4 | 3 | ancoms 460 | 1 ⊢ ((𝜓 ∧ 𝜑) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-cleq 2725 |
This theorem is referenced by: fmptco 7079 axcontlem4 27965 usgredg4 28214 cusgrsize 28451 uspgr2wlkeqi 28645 clwwlkf1 29042 eigorthi 30828 goeleq12bg 34007 expdiophlem2 41393 pwssplit4 41463 fcoresf1 45393 prproropf1olem4 45788 fmtnoodd 45815 |
Copyright terms: Public domain | W3C validator |