Theorem dbitrns-P4.16.RC 429

Description: Inference Form of dbitrns-P4.16 428. †

Hypotheses

Ref	Expression
dbitrns-P4.16.RC.1	⊢ (𝜑 ↔ 𝜓)
dbitrns-P4.16.RC.2	⊢ (𝜓 ↔ 𝜒)
dbitrns-P4.16.RC.3	⊢ (𝜒 ↔ 𝜗)

Assertion

Ref	Expression
dbitrns-P4.16.RC	⊢ (𝜑 ↔ 𝜗)

Proof of Theorem dbitrns-P4.16.RC

Step	Hyp	Ref	Expression
1		dbitrns-P4.16.RC.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
2	1	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓))
3		dbitrns-P4.16.RC.2	. . . 4 ⊢ (𝜓 ↔ 𝜒)
4	3	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜓 ↔ 𝜒))
5		dbitrns-P4.16.RC.3	. . . 4 ⊢ (𝜒 ↔ 𝜗)
6	5	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜒 ↔ 𝜗))
7	2, 4, 6	dbitrns-P4.16 428	. 2 ⊢ (⊤ → (𝜑 ↔ 𝜗))
8	7	ndtruee-P3.18 183	1 ⊢ (𝜑 ↔ 𝜗)

Syntax hints:   ↔ wff-bi 104  ⊤wff-true 153

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:  oroverbi-P4.28b  469  imoverbi-P4.30b  479  biasandor-P4.34a  491  cbvexv-P5  660  nfrleq-P6  687  cbvex-P6  752  qcexandr-P6  761  qcexandl-P6  762  psubleq-P6  783  psubnfr-P6  784  psubneg-P6  788  psuband-P6  792  psubspliteq-P6  800  dfpsub-P7  978