Theorem lemma-L7.03 962

Description: Proper Substitution Applied To ENF Variable Lemma. †

'𝑥' cannot occur in '𝑡'.

Hypotheses

Ref	Expression
lemma-L7.03.1	⊢ Ⅎ𝑥𝛾
lemma-L7.03.2	⊢ (𝛾 → Ⅎ𝑥𝜑)

Assertion

Ref	Expression
lemma-L7.03	⊢ (𝛾 → ([𝑡 / 𝑥]𝜑 ↔ 𝜑))

Distinct variable group:   𝑡,𝑥

Proof of Theorem lemma-L7.03

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lemma-L7.03.1	. . 3 ⊢ Ⅎ𝑥𝛾
2		ndnfrv-P7.1 826	. . 3 ⊢ Ⅎ𝑥 𝑦 = 𝑡
3		lemma-L7.03.2	. . . 4 ⊢ (𝛾 → Ⅎ𝑥𝜑)
4		ndpsub3-P7.15 840	. . . . 5 ⊢ Ⅎ𝑥[𝑡 / 𝑥]𝜑
5	4	rcp-NDIMP0addall 207	. . . 4 ⊢ (𝛾 → Ⅎ𝑥[𝑡 / 𝑥]𝜑)
6	3, 5	ndnfrbi-P7.6 831	. . 3 ⊢ (𝛾 → Ⅎ𝑥(𝜑 ↔ [𝑡 / 𝑥]𝜑))
7		ndpsub2-P7.14 839	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝜑 ↔ [𝑡 / 𝑥]𝜑))
8		lemma-L7.01a 924	. . . . . 6 ⊢ ([𝑥 / 𝑦] 𝑦 = 𝑡 ↔ 𝑥 = 𝑡)
9	8	bisym-P3.33b.RC 299	. . . . 5 ⊢ (𝑥 = 𝑡 ↔ [𝑥 / 𝑦] 𝑦 = 𝑡)
10	7, 9	subiml2-P4.RC 541	. . . 4 ⊢ ([𝑥 / 𝑦] 𝑦 = 𝑡 → (𝜑 ↔ [𝑡 / 𝑥]𝜑))
11	10	rcp-NDIMP0addall 207	. . 3 ⊢ (𝛾 → ([𝑥 / 𝑦] 𝑦 = 𝑡 → (𝜑 ↔ [𝑡 / 𝑥]𝜑)))
12		axL6ex-P7 925	. . . 4 ⊢ ∃𝑦 𝑦 = 𝑡
13	12	rcp-NDIMP0addall 207	. . 3 ⊢ (𝛾 → ∃𝑦 𝑦 = 𝑡)
14	1, 2, 6, 11, 13	ndexe-P7.20 845	. 2 ⊢ (𝛾 → (𝜑 ↔ [𝑡 / 𝑥]𝜑))
15	14	bisym-P3.33b 298	1 ⊢ (𝛾 → ([𝑡 / 𝑥]𝜑 ↔ 𝜑))

Syntax hints:  term-obj 1   = wff-equals 6   → wff-imp 10   ↔ wff-bi 104  ∃wff-exists 595  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-L10 27  ax-L11 28  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:  nfrgencl-P7  965