Theorem psubnfrv-P7 927

Description: Proper Substitution Applied To ENF Variable (variable restriction). †

'𝑥' cannot occur in '𝑡'.

Hypothesis

Ref	Expression
psubnfrv-P7.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
psubnfrv-P7	⊢ ([𝑡 / 𝑥]𝜑 ↔ 𝜑)

Distinct variable group:   𝑡,𝑥

Proof of Theorem psubnfrv-P7

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		psubnfrv-P7.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		ndpsub3-P7.15 840	. . . 4 ⊢ Ⅎ𝑥[𝑡 / 𝑥]𝜑
3	1, 2	ndnfrbi-P7.6.RC 879	. . 3 ⊢ Ⅎ𝑥(𝜑 ↔ [𝑡 / 𝑥]𝜑)
4		ndpsub2-P7.14 839	. . . 4 ⊢ (𝑥 = 𝑡 → (𝜑 ↔ [𝑡 / 𝑥]𝜑))
5		lemma-L7.01a 924	. . . . 5 ⊢ ([𝑥 / 𝑦] 𝑦 = 𝑡 ↔ 𝑥 = 𝑡)
6	5	bisym-P3.33b.RC 299	. . . 4 ⊢ (𝑥 = 𝑡 ↔ [𝑥 / 𝑦] 𝑦 = 𝑡)
7	4, 6	subiml2-P4.RC 541	. . 3 ⊢ ([𝑥 / 𝑦] 𝑦 = 𝑡 → (𝜑 ↔ [𝑡 / 𝑥]𝜑))
8		axL6ex-P7 925	. . 3 ⊢ ∃𝑦 𝑦 = 𝑡
9	3, 7, 8	ndexew-P7.RC.VR1of2 888	. 2 ⊢ (𝜑 ↔ [𝑡 / 𝑥]𝜑)
10	9	bisym-P3.33b.RC 299	1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:  term-obj 1   = 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-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:  nfrgen-P7  928  nfrexgen-P7  931