Theorem psuband-P6 792

Description: Proper Substitution Over Conjunction.

Assertion

Ref	Expression
psuband-P6	⊢ ([𝑡 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑡 / 𝑥]𝜑 ∧ [𝑡 / 𝑥]𝜓))

Proof of Theorem psuband-P6

Step	Hyp	Ref	Expression
1		andasim-P3.46a 356	. . . 4 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
2	1	psubleq-P6 783	. . 3 ⊢ ([𝑡 / 𝑥](𝜑 ∧ 𝜓) ↔ [𝑡 / 𝑥] ¬ (𝜑 → ¬ 𝜓))
3		psubneg-P6 788	. . 3 ⊢ ([𝑡 / 𝑥] ¬ (𝜑 → ¬ 𝜓) ↔ ¬ [𝑡 / 𝑥](𝜑 → ¬ 𝜓))
4		psubim-P6 791	. . . . 5 ⊢ ([𝑡 / 𝑥](𝜑 → ¬ 𝜓) ↔ ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥] ¬ 𝜓))
5		psubneg-P6 788	. . . . . 6 ⊢ ([𝑡 / 𝑥] ¬ 𝜓 ↔ ¬ [𝑡 / 𝑥]𝜓)
6	5	subimr-P3.40b.RC 328	. . . . 5 ⊢ (([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥] ¬ 𝜓) ↔ ([𝑡 / 𝑥]𝜑 → ¬ [𝑡 / 𝑥]𝜓))
7	4, 6	bitrns-P3.33c.RC 303	. . . 4 ⊢ ([𝑡 / 𝑥](𝜑 → ¬ 𝜓) ↔ ([𝑡 / 𝑥]𝜑 → ¬ [𝑡 / 𝑥]𝜓))
8	7	subneg-P3.39.RC 324	. . 3 ⊢ (¬ [𝑡 / 𝑥](𝜑 → ¬ 𝜓) ↔ ¬ ([𝑡 / 𝑥]𝜑 → ¬ [𝑡 / 𝑥]𝜓))
9	2, 3, 8	dbitrns-P4.16.RC 429	. 2 ⊢ ([𝑡 / 𝑥](𝜑 ∧ 𝜓) ↔ ¬ ([𝑡 / 𝑥]𝜑 → ¬ [𝑡 / 𝑥]𝜓))
10		andasim-P3.46a 356	. . 3 ⊢ (([𝑡 / 𝑥]𝜑 ∧ [𝑡 / 𝑥]𝜓) ↔ ¬ ([𝑡 / 𝑥]𝜑 → ¬ [𝑡 / 𝑥]𝜓))
11	10	bisym-P3.33b.RC 299	. 2 ⊢ (¬ ([𝑡 / 𝑥]𝜑 → ¬ [𝑡 / 𝑥]𝜓) ↔ ([𝑡 / 𝑥]𝜑 ∧ [𝑡 / 𝑥]𝜓))
12	9, 11	bitrns-P3.33c.RC 303	1 ⊢ ([𝑡 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑡 / 𝑥]𝜑 ∧ [𝑡 / 𝑥]𝜓))

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ↔ wff-bi 104   ∧ wff-and 132  [wff-psub 714

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16  ax-L5 17  ax-L6 18  ax-L7 19  ax-L10 27  ax-L12 29

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682  df-psub-D6.2 716

This theorem is referenced by:  psubspliteq-P6  800  psubsplitelof-P6  801