Theorem subbid-P3.41c 336

Description: Dual Substitution Law for '↔'. †

Hypotheses

Ref	Expression
subbid-P3.41c.1	⊢ (𝛾 → (𝜑 ↔ 𝜓))
subbid-P3.41c.2	⊢ (𝛾 → (𝜒 ↔ 𝜗))

Assertion

Ref	Expression
subbid-P3.41c	⊢ (𝛾 → ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜗)))

Proof of Theorem subbid-P3.41c

Step	Hyp	Ref	Expression
1		subbid-P3.41c.1	. . 3 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
2	1	subbil-P3.41a 332	. 2 ⊢ (𝛾 → ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒)))
3		subbid-P3.41c.2	. . 3 ⊢ (𝛾 → (𝜒 ↔ 𝜗))
4	3	subbir-P3.41b 334	. 2 ⊢ (𝛾 → ((𝜓 ↔ 𝜒) ↔ (𝜓 ↔ 𝜗)))
5	2, 4	bitrns-P3.33c 302	1 ⊢ (𝛾 → ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜗)))

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:  subbid-P3.41c.RC  337  negbicancel-P4.11  419  negbicancelint-P4.14  424  subbid2-P4  550  example-E5.03a  665