Theorem suborl-P3.43a 346

Description: Left Substitution Theorem for '∨' . †

Hypothesis

Ref	Expression
suborl-P3.43a.1	⊢ (𝛾 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
suborl-P3.43a	⊢ (𝛾 → ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒)))

Proof of Theorem suborl-P3.43a

Step	Hyp	Ref	Expression
1		suborl-P3.43a.1	. . 3 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
2	1	suborl-P3.43a-L1 345	. 2 ⊢ (𝛾 → ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜒)))
3	1	bisym-P3.33b 298	. . 3 ⊢ (𝛾 → (𝜓 ↔ 𝜑))
4	3	suborl-P3.43a-L1 345	. 2 ⊢ (𝛾 → ((𝜓 ∨ 𝜒) → (𝜑 ∨ 𝜒)))
5	2, 4	ndbii-P3.13 178	1 ⊢ (𝛾 → ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒)))

Syntax hints:   → wff-imp 10   ↔ wff-bi 104   ∨ wff-or 144

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-rcp-AND3 161

This theorem is referenced by:  suborl-P3.43a.RC  347  suborr-P3.43b  348  subord-P3.43c  350  suborl2-P4  558