Theorem subandr-P3.42b 341

Description: Right Substitution Law for '∧'. †

Hypothesis

Ref	Expression
subandr-P3.42b.1	⊢ (𝛾 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
subandr-P3.42b	⊢ (𝛾 → ((𝜒 ∧ 𝜑) ↔ (𝜒 ∧ 𝜓)))

Proof of Theorem subandr-P3.42b

Step	Hyp	Ref	Expression
1		andcomm-P3.35 314	. . . 4 ⊢ ((𝜒 ∧ 𝜑) ↔ (𝜑 ∧ 𝜒))
2	1	rcp-NDIMP0addall 207	. . 3 ⊢ (𝛾 → ((𝜒 ∧ 𝜑) ↔ (𝜑 ∧ 𝜒)))
3		subandr-P3.42b.1	. . . 4 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
4	3	subandl-P3.42a 339	. . 3 ⊢ (𝛾 → ((𝜑 ∧ 𝜒) ↔ (𝜓 ∧ 𝜒)))
5	2, 4	bitrns-P3.33c 302	. 2 ⊢ (𝛾 → ((𝜒 ∧ 𝜑) ↔ (𝜓 ∧ 𝜒)))
6		andcomm-P3.35 314	. . 3 ⊢ ((𝜓 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓))
7	6	rcp-NDIMP0addall 207	. 2 ⊢ (𝛾 → ((𝜓 ∧ 𝜒) ↔ (𝜒 ∧ 𝜓)))
8	5, 7	bitrns-P3.33c 302	1 ⊢ (𝛾 → ((𝜒 ∧ 𝜑) ↔ (𝜒 ∧ 𝜓)))

Syntax hints:   → wff-imp 10   ↔ wff-bi 104   ∧ wff-and 132

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:  subandr-P3.42b.RC  342  subandd-P3.42c  343  subandr2-P4  554