Theorem subneg-P3.39 323

Description: Substitution Law for '¬'. †

Note that the proof of this theorem uses trnsp-P3.31c 285, which does not rely on the Law of Excluded Middle. This means this law is deducible with intuitionist logic.

Hypothesis

Ref	Expression
subneg-P3.39.1	⊢ (𝛾 → (𝜑 ↔ 𝜓))

Assertion

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

Proof of Theorem subneg-P3.39

Step	Hyp	Ref	Expression
1		subneg-P3.39.1	. . . 4 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
2	1	ndbier-P3.15 180	. . 3 ⊢ (𝛾 → (𝜓 → 𝜑))
3	2	trnsp-P3.31c 285	. 2 ⊢ (𝛾 → (¬ 𝜑 → ¬ 𝜓))
4	1	ndbief-P3.14 179	. . 3 ⊢ (𝛾 → (𝜑 → 𝜓))
5	4	trnsp-P3.31c 285	. 2 ⊢ (𝛾 → (¬ 𝜓 → ¬ 𝜑))
6	3, 5	ndbii-P3.13 178	1 ⊢ (𝛾 → (¬ 𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wff-neg 9   → 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-rcp-AND3 161

This theorem is referenced by:  subneg-P3.39.RC  324  negbicancel-P4.11  419  negbicancelint-P4.14  424  subneg2-P4  538  exiisub-P5  655  cbvexv-P5  660  exiw-P5  662  lemma-L5.05a  668  gennallw-P6  678  gennexw-P6  679  nfrex2w-P6  695  nfrexgenw-P6  696  exgennfrw-P6  697  cbvex-P6  752