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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 2838 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
4 | 3 | ancoms 461 | 1 ⊢ ((𝜓 ∧ 𝜑) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-9 2120 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-cleq 2814 |
This theorem is referenced by: fmptco 6885 axcontlem4 26747 usgredg4 26993 cusgrsize 27230 uspgr2wlkeqi 27423 clwwlkf1 27822 eigorthi 29608 goeleq12bg 32591 expdiophlem2 39612 pwssplit4 39682 prproropf1olem4 43662 fmtnoodd 43689 |
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