Theorem subimr2-P4.RC 543

Description: Inference Form of subimr2-P4 542. †

Hypotheses

Ref	Expression
subimr2-P4.RC.1	⊢ (𝜒 → 𝜑)
subimr2-P4.RC.2	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
subimr2-P4.RC	⊢ (𝜒 → 𝜓)

Proof of Theorem subimr2-P4.RC

Step	Hyp	Ref	Expression
1		subimr2-P4.RC.1	. . . 4 ⊢ (𝜒 → 𝜑)
2	1	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜒 → 𝜑))
3		subimr2-P4.RC.2	. . . 4 ⊢ (𝜑 ↔ 𝜓)
4	3	ndtruei-P3.17 182	. . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓))
5	2, 4	subimr2-P4 542	. 2 ⊢ (⊤ → (𝜒 → 𝜓))
6	5	ndtruee-P3.18 183	1 ⊢ (𝜒 → 𝜓)

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:  andassoc2a-P4  568  andassoc2b-P4  570  orassoc2a-P4  572  orassoc2b-P4  574  subeqr-P5-L1  634  exiw-P5  662  example-E6.01a  706  solvesub-P6b  707  example-E6.02a  712  solvedsub-P6b  713  specpsub-P6  721  psubtoisubv-P6  725  psubtoisub-P6  765  lemma-L6.07a-L1  770  psubim-P6-L1  789  gennfrcl-L6  812  exgennfrcl-L6  814  nfrimd-P6  815  nfrandd-P6  816  nfrord-P6  817  nfrbid-P6  818  nfrall2d-P6  819  nfrex2d-P6  820  ndalli-P6  822  qremexd-P6  823  nfrgen-P7  928  alle-P7  941  exi-P7  951  qimeqex-P7-L1  1054  example-E7.1b  1075