Theorem tbitrns-P4.17.RC 431

Description: Inference Form of tbitrns-P4.17 430. †

Hypotheses

Ref	Expression
tbitrns-P4.17.RC.1	⊢ (𝜑 ↔ 𝜓)
tbitrns-P4.17.RC.2	⊢ (𝜓 ↔ 𝜒)
tbitrns-P4.17.RC.3	⊢ (𝜒 ↔ 𝜗)
tbitrns-P4.17.RC.4	⊢ (𝜗 ↔ 𝜏)

Assertion

Ref	Expression
tbitrns-P4.17.RC	⊢ (𝜑 ↔ 𝜏)

Proof of Theorem tbitrns-P4.17.RC

Step	Hyp	Ref	Expression
1		tbitrns-P4.17.RC.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
2	1	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓))
3		tbitrns-P4.17.RC.2	. . . 4 ⊢ (𝜓 ↔ 𝜒)
4	3	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜓 ↔ 𝜒))
5		tbitrns-P4.17.RC.3	. . . 4 ⊢ (𝜒 ↔ 𝜗)
6	5	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜒 ↔ 𝜗))
7		tbitrns-P4.17.RC.4	. . . 4 ⊢ (𝜗 ↔ 𝜏)
8	7	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜗 ↔ 𝜏))
9	2, 4, 6, 8	tbitrns-P4.17 430	. 2 ⊢ (⊤ → (𝜑 ↔ 𝜏))
10	9	ndtruee-P3.18 183	1 ⊢ (𝜑 ↔ 𝜏)

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:  biasandor-P4.34a  491  dfnfreealt-P6  683  nfrneg-P6  688  solvesub-P6a  704  lemma-L6.02a  726  psuball2v-P6  796  psubex2v-P6  797  psubsplitelof-P6  801  dfnfree-P7  968