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Metamath Proof Explorer

Theorem dedth2v

Description: Weak deduction theorem for eliminating a hypothesis with 2 class variables. Note: if the hypothesis can be separated into two hypotheses, each with one class variable, then dedth2h is simpler to use. See also comments in dedth . (Contributed by NM, 13-Aug-1999) (Proof shortened by Eric Schmidt, 28-Jul-2009)

	Ref	Expression
Hypotheses	dedth2v.1	⊢ ( 𝐴 = if ( 𝜑 , 𝐴 , 𝐶 ) → ( 𝜓 ↔ 𝜒 ) )
	dedth2v.2	⊢ ( 𝐵 = if ( 𝜑 , 𝐵 , 𝐷 ) → ( 𝜒 ↔ 𝜃 ) )
	dedth2v.3	⊢ 𝜃
Assertion	dedth2v	⊢ ( 𝜑 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		dedth2v.1	⊢ ( 𝐴 = if ( 𝜑 , 𝐴 , 𝐶 ) → ( 𝜓 ↔ 𝜒 ) )
2		dedth2v.2	⊢ ( 𝐵 = if ( 𝜑 , 𝐵 , 𝐷 ) → ( 𝜒 ↔ 𝜃 ) )
3		dedth2v.3	⊢ 𝜃
4	1 2 3	dedth2h	⊢ ( ( 𝜑 ∧ 𝜑 ) → 𝜓 )
5	4	anidms	⊢ ( 𝜑 → 𝜓 )