Theorem psubsuccv-P6 806

Description: Proper Substitution Over Successor Function (variable restriction).

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

Hypothesis

Ref	Expression
psubsuccv-P6.1	⊢ ([𝑤 / 𝑥] 𝑎 = 𝑡 ↔ 𝑎 = 𝑢)

Assertion

Ref	Expression
psubsuccv-P6	⊢ ([𝑤 / 𝑥] 𝑏 = s‘𝑡 ↔ 𝑏 = s‘𝑢)

Distinct variable groups:   𝑡,𝑎   𝑢,𝑎   𝑤,𝑎   𝑡,𝑏   𝑢,𝑏   𝑤,𝑏,𝑥,𝑎

Proof of Theorem psubsuccv-P6

Step	Hyp	Ref	Expression
1		lemma-L6.06a 766	. . . . 5 ⊢ Ⅎ𝑥[𝑤 / 𝑥] 𝑎 = 𝑡
2		psubsuccv-P6.1	. . . . . 6 ⊢ ([𝑤 / 𝑥] 𝑎 = 𝑡 ↔ 𝑎 = 𝑢)
3	2	nfrleq-P6 687	. . . . 5 ⊢ (Ⅎ𝑥[𝑤 / 𝑥] 𝑎 = 𝑡 ↔ Ⅎ𝑥 𝑎 = 𝑢)
4	1, 3	bimpf-P4.RC 532	. . . 4 ⊢ Ⅎ𝑥 𝑎 = 𝑢
5	4	nfrterm-P6 779	. . 3 ⊢ Ⅎ𝑥 𝑏 = 𝑢
6	5	nfrsucc-P6 780	. 2 ⊢ Ⅎ𝑥 𝑏 = s‘𝑢
7	2	psubsuccv-P6-L1 805	. 2 ⊢ (𝑥 = 𝑤 → (𝑏 = s‘𝑡 ↔ 𝑏 = s‘𝑢))
8	6, 7	isubtopsubv-P6 727	1 ⊢ ([𝑤 / 𝑥] 𝑏 = s‘𝑡 ↔ 𝑏 = s‘𝑢)

Syntax hints:  term-obj 1  s‘term_succ 3   = 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-succ 22  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)