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Metamath Proof Explorer

Theorem eqeqan12rd

Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994)

Ref Expression
Hypotheses eqeqan12rd.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
eqeqan12rd.2 ⊢ ( 𝜓 → 𝐶 = 𝐷 )
Assertion eqeqan12rd ( ( 𝜓 ∧ 𝜑 ) → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐷 ) )

Proof

Step Hyp Ref Expression
1 eqeqan12rd.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
2 eqeqan12rd.2 ⊢ ( 𝜓 → 𝐶 = 𝐷 )
3 1 2 eqeqan12d ⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐷 ) )
4 3 ancoms ⊢ ( ( 𝜓 ∧ 𝜑 ) → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐷 ) )