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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 2752 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
4 | 3 | ancoms 462 | 1 ⊢ ((𝜓 ∧ 𝜑) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = 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 2121 ax-ext 2709 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-cleq 2730 |
This theorem is referenced by: fmptco 6962 axcontlem4 27082 usgredg4 27329 cusgrsize 27566 uspgr2wlkeqi 27759 clwwlkf1 28156 eigorthi 29942 goeleq12bg 33047 expdiophlem2 40575 pwssplit4 40645 fcoresf1 44263 prproropf1olem4 44659 fmtnoodd 44686 |
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