Theorem subiml2-P4.RC 541

Description: Inference Form of subiml2-P4 540. †

Hypotheses

Ref	Expression
subiml2-P4.RC.1	⊢ (𝜑 → 𝜒)
subiml2-P4.RC.2	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
subiml2-P4.RC	⊢ (𝜓 → 𝜒)

Proof of Theorem subiml2-P4.RC

Step	Hyp	Ref	Expression
1		subiml2-P4.RC.1	. . . 4 ⊢ (𝜑 → 𝜒)
2	1	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜑 → 𝜒))
3		subiml2-P4.RC.2	. . . 4 ⊢ (𝜑 ↔ 𝜓)
4	3	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓))
5	2, 4	subiml2-P4 540	. 2 ⊢ (⊤ → (𝜓 → 𝜒))
6	5	ndtruee-P3.18 183	1 ⊢ (𝜓 → 𝜒)

Syntax hints:   → wff-imp 10   ↔ 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:  qimeqallhalf-P5  609  qimeqex-P5-L1  610  eqtrns-P5  630  subeqr-P5  635  subelofl-P5  638  subelofr-P5  640  exiisub-P5  655  cbvallv-P5  659  specw-P5  661  lemma-L5.04a  667  example-E5.04a  675  spec-P6  719  cbvall-P6  751  lemma-L6.07a-L2  771  psubim-P6-L1  789  psubim-P6-L2  790  psubnfrv-P7  927  psubinv-P7  939  lemma-L7.02a-L1  943  lemma-L7.03  962  cbvall-P7  1061  cbvex-P7  1066