Theorem subneg-P2.7 127

Description: Substitution Law for '¬'.

Assertion

Ref	Expression
subneg-P2.7	⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓))

Proof of Theorem subneg-P2.7

Step	Hyp	Ref	Expression
1		birev-P2.5b 115	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
2	1	trnsp-P1.15c.AC.SH 82	. 2 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 → ¬ 𝜓))
3		bifwd-P2.5a 111	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
4	3	trnsp-P1.15c.AC.SH 82	. 2 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜓 → ¬ 𝜑))
5	2, 4	bicmb-P2.5c.AC.2SH 121	1 ⊢ ((𝜑 ↔ 𝜓) → (¬ 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

This theorem is referenced by: (None)