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Metamath Proof Explorer

Theorem orim12dALT

Description: Alternate proof of orim12d which does not depend on df-an . This is an illustration of the conservativity of definitions (definitions do not permit to prove additional theorems whose statements do not contain the defined symbol). (Contributed by Wolf Lammen, 8-Aug-2022) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	orim12dALT.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
	orim12dALT.2	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
Assertion	orim12dALT	⊢ ( 𝜑 → ( ( 𝜓 ∨ 𝜃 ) → ( 𝜒 ∨ 𝜏 ) ) )

Proof

Step	Hyp	Ref	Expression
1		orim12dALT.1	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2		orim12dALT.2	⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
3		pm2.53	⊢ ( ( 𝜓 ∨ 𝜃 ) → ( ¬ 𝜓 → 𝜃 ) )
4	1	con3d	⊢ ( 𝜑 → ( ¬ 𝜒 → ¬ 𝜓 ) )
5	4 2	imim12d	⊢ ( 𝜑 → ( ( ¬ 𝜓 → 𝜃 ) → ( ¬ 𝜒 → 𝜏 ) ) )
6		pm2.54	⊢ ( ( ¬ 𝜒 → 𝜏 ) → ( 𝜒 ∨ 𝜏 ) )
7	3 5 6	syl56	⊢ ( 𝜑 → ( ( 𝜓 ∨ 𝜃 ) → ( 𝜒 ∨ 𝜏 ) ) )