Theorem bisym-P3.33b 298

Description: Equivalence Property: '↔' Symmetry. †

Hypothesis

Ref	Expression
bisym-P3.33b.1	⊢ (𝛾 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
bisym-P3.33b	⊢ (𝛾 → (𝜓 ↔ 𝜑))

Proof of Theorem bisym-P3.33b

Step	Hyp	Ref	Expression
1		bisym-P3.33b.1	. . 3 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
2	1	ndbier-P3.15 180	. 2 ⊢ (𝛾 → (𝜓 → 𝜑))
3	1	ndbief-P3.14 179	. 2 ⊢ (𝛾 → (𝜑 → 𝜓))
4	2, 3	ndbii-P3.13 178	1 ⊢ (𝛾 → (𝜓 ↔ 𝜑))

Syntax hints:   → 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:  bisym-P3.33b.RC  299  bisym-P3.33b.CL  300  subbil-P3.41a-L1  331  subbil-P3.41a  332  subandl-P3.42a  339  suborl-P3.43a  346  lemma-L5.04a  667  lemma-L7.03  962  cbvall-P7  1061  cbvex-P7  1066