Theorem psubmultv-P6 810

Description: Proper Substitution Over Multiplication (variable restriction).

'𝑎', '𝑏', and '𝑐' are distinct from all other variables and '𝑥' cannot occur in '𝑤'.

Hypotheses

Ref	Expression
psubmultv-P6.1	⊢ ([𝑤 / 𝑥] 𝑎 = 𝑡₁ ↔ 𝑎 = 𝑢₁)
psubmultv-P6.2	⊢ ([𝑤 / 𝑥] 𝑏 = 𝑡₂ ↔ 𝑏 = 𝑢₂)

Assertion

Ref	Expression
psubmultv-P6	⊢ ([𝑤 / 𝑥] 𝑐 = (𝑡₁ ⋅ 𝑡₂) ↔ 𝑐 = (𝑢₁ ⋅ 𝑢₂))

Distinct variable groups:   𝑡₁,𝑎   𝑡₂,𝑎   𝑢₁,𝑎   𝑢₂,𝑎   𝑤,𝑎   𝑡₁,𝑏   𝑡₂,𝑏   𝑢₁,𝑏   𝑢₂,𝑏   𝑤,𝑏   𝑡₁,𝑐   𝑡₂,𝑐   𝑢₁,𝑐   𝑢₂,𝑐   𝑤,𝑐,𝑥   𝑎,𝑏,𝑐,𝑥

Proof of Theorem psubmultv-P6

Step	Hyp	Ref	Expression
1		lemma-L6.06a 766	. . . . 5 ⊢ Ⅎ𝑥[𝑤 / 𝑥] 𝑎 = 𝑡₁
2		psubmultv-P6.1	. . . . . 6 ⊢ ([𝑤 / 𝑥] 𝑎 = 𝑡₁ ↔ 𝑎 = 𝑢₁)
3	2	nfrleq-P6 687	. . . . 5 ⊢ (Ⅎ𝑥[𝑤 / 𝑥] 𝑎 = 𝑡₁ ↔ Ⅎ𝑥 𝑎 = 𝑢₁)
4	1, 3	bimpf-P4.RC 532	. . . 4 ⊢ Ⅎ𝑥 𝑎 = 𝑢₁
5	4	nfrterm-P6 779	. . 3 ⊢ Ⅎ𝑥 𝑐 = 𝑢₁
6		lemma-L6.06a 766	. . . . 5 ⊢ Ⅎ𝑥[𝑤 / 𝑥] 𝑏 = 𝑡₂
7		psubmultv-P6.2	. . . . . 6 ⊢ ([𝑤 / 𝑥] 𝑏 = 𝑡₂ ↔ 𝑏 = 𝑢₂)
8	7	nfrleq-P6 687	. . . . 5 ⊢ (Ⅎ𝑥[𝑤 / 𝑥] 𝑏 = 𝑡₂ ↔ Ⅎ𝑥 𝑏 = 𝑢₂)
9	6, 8	bimpf-P4.RC 532	. . . 4 ⊢ Ⅎ𝑥 𝑏 = 𝑢₂
10	9	nfrterm-P6 779	. . 3 ⊢ Ⅎ𝑥 𝑐 = 𝑢₂
11	5, 10	nfrmult-P6 782	. 2 ⊢ Ⅎ𝑥 𝑐 = (𝑢₁ ⋅ 𝑢₂)
12	2, 7	psubmultv-P6-L1 809	. 2 ⊢ (𝑥 = 𝑤 → (𝑐 = (𝑡₁ ⋅ 𝑡₂) ↔ 𝑐 = (𝑢₁ ⋅ 𝑢₂)))
13	11, 12	isubtopsubv-P6 727	1 ⊢ ([𝑤 / 𝑥] 𝑐 = (𝑡₁ ⋅ 𝑡₂) ↔ 𝑐 = (𝑢₁ ⋅ 𝑢₂))

Syntax hints:  term-obj 1   ⋅ term-mult 5   = wff-equals 6   ↔ wff-bi 104  Ⅎwff-nfree 681  [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-L9-multl 25  ax-L9-multr 26  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: (None)