Theorem subbil2-P4.RC 547

Description: Inference Form of subbil2-P4 546. †

Hypotheses

Ref	Expression
subbil2-P4.RC.1	⊢ (𝜑 ↔ 𝜒)
subbil2-P4.RC.2	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
subbil2-P4.RC	⊢ (𝜓 ↔ 𝜒)

Proof of Theorem subbil2-P4.RC

Step	Hyp	Ref	Expression
1		subbil2-P4.RC.1	. . . 4 ⊢ (𝜑 ↔ 𝜒)
2	1	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜑 ↔ 𝜒))
3		subbil2-P4.RC.2	. . . 4 ⊢ (𝜑 ↔ 𝜓)
4	3	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓))
5	2, 4	subbil2-P4 546	. 2 ⊢ (⊤ → (𝜓 ↔ 𝜒))
6	5	ndtruee-P3.18 183	1 ⊢ (𝜓 ↔ 𝜒)

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:  qcallimrv-P5  671  qceximrv-P5  672  qcallimlv-P5  673  qceximlv-P5  674  qcallimr-P6  757  qceximr-P6  758  qcalliml-P6  759  qceximl-P6  760  dfexists-P7  959  qcallimr-P8  1122  qceximr-P8  1123  qcalliml-P8  1124  qceximl-P8  1125