Theorem subbir-P3.41b 334

Description: Right Substitution Law for '↔'. †

Hypothesis

Ref	Expression
subbir-P3.41b.1	⊢ (𝛾 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
subbir-P3.41b	⊢ (𝛾 → ((𝜒 ↔ 𝜑) ↔ (𝜒 ↔ 𝜓)))

Proof of Theorem subbir-P3.41b

Step	Hyp	Ref	Expression
1		bisym-P3.33b.CL.SYM 301	. . . 4 ⊢ ((𝜒 ↔ 𝜑) ↔ (𝜑 ↔ 𝜒))
2	1	rcp-NDIMP0addall 207	. . 3 ⊢ (𝛾 → ((𝜒 ↔ 𝜑) ↔ (𝜑 ↔ 𝜒)))
3		subbir-P3.41b.1	. . . 4 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
4	3	subbil-P3.41a 332	. . 3 ⊢ (𝛾 → ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒)))
5	2, 4	bitrns-P3.33c 302	. 2 ⊢ (𝛾 → ((𝜒 ↔ 𝜑) ↔ (𝜓 ↔ 𝜒)))
6		bisym-P3.33b.CL.SYM 301	. . 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

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-true-D2.4 155

This theorem is referenced by:  subbir-P3.41b.RC  335  subbid-P3.41c  336  subbir2-P4  548  trnsvsubw-P6  710