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Mirrors > Home > ILE Home > Th. List > eqeqan12rd | 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 2052 | . 2 ⊢ ((φ ∧ ψ) → (A = 𝐶 ↔ B = 𝐷)) |
4 | 3 | ancoms 255 | 1 ⊢ ((ψ ∧ φ) → (A = 𝐶 ↔ B = 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-gen 1335 ax-4 1397 ax-17 1416 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-cleq 2030 |
This theorem is referenced by: (None) |
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