Theorem subneg-P3.39.RC 324

Description: Inference Form of subneg-P3.39 323. †

Hypothesis

Ref	Expression
subneg-P3.39.RC.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
subneg-P3.39.RC	⊢ (¬ 𝜑 ↔ ¬ 𝜓)

Proof of Theorem subneg-P3.39.RC

Step	Hyp	Ref	Expression
1		subneg-P3.39.RC.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
2	1	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓))
3	2	subneg-P3.39 323	. 2 ⊢ (⊤ → (¬ 𝜑 ↔ ¬ 𝜓))
4	3	ndtruee-P3.18 183	1 ⊢ (¬ 𝜑 ↔ ¬ 𝜓)

Syntax hints:  ¬ wff-neg 9   ↔ 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  df-rcp-AND3 161

This theorem is referenced by:  allnegex-P5  597  exnegall-P5  598  allasex-P5  599  cbvexv-P5  660  nfrleq-P6  687  cbvex-P6  752  qcexandr-P6  761  qcexandl-P6  762  psubneg-P6-L1  787  psubneg-P6  788  psuband-P6  792  psubex2v-P6  797  dfexists-P7  959  dfexistsint-P7  960  exnegall-P7  1046  allasex-P7  1048