Theorem suborr-P3.43b 348

Description: Right Substitution Law for '∨' . †

Hypothesis

Ref	Expression
suborr-P3.43b.1	⊢ (𝛾 → (𝜑 ↔ 𝜓))

Assertion

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

Proof of Theorem suborr-P3.43b

Step	Hyp	Ref	Expression
1		orcomm-P3.37 319	. . . 4 ⊢ ((𝜒 ∨ 𝜑) ↔ (𝜑 ∨ 𝜒))
2	1	rcp-NDIMP0addall 207	. . 3 ⊢ (𝛾 → ((𝜒 ∨ 𝜑) ↔ (𝜑 ∨ 𝜒)))
3		suborr-P3.43b.1	. . . 4 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
4	3	suborl-P3.43a 346	. . 3 ⊢ (𝛾 → ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒)))
5	2, 4	bitrns-P3.33c 302	. 2 ⊢ (𝛾 → ((𝜒 ∨ 𝜑) ↔ (𝜓 ∨ 𝜒)))
6		orcomm-P3.37 319	. . 3 ⊢ ((𝜓 ∨ 𝜒) ↔ (𝜒 ∨ 𝜓))
7	6	rcp-NDIMP0addall 207	. 2 ⊢ (𝛾 → ((𝜓 ∨ 𝜒) ↔ (𝜒 ∨ 𝜓)))
8	5, 7	bitrns-P3.33c 302	1 ⊢ (𝛾 → ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓)))

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-true-D2.4 155  df-rcp-AND3 161

This theorem is referenced by:  suborr-P3.43b.RC  349  subord-P3.43c  350  suborr2-P4  560