Theorem psubinv-P7 939

Description: Proper Substitution Inverse Property. †

Hypothesis

Ref	Expression
psubinv-P7.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
psubinv-P7	⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ 𝜑)

Distinct variable group:   𝑥,𝑦

Proof of Theorem psubinv-P7

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		psubinv-P7.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2		ndpsub3-P7.15 840	. . . 4 ⊢ Ⅎ𝑦[𝑥 / 𝑦][𝑦 / 𝑥]𝜑
3	1, 2	ndnfrbi-P7.6.RC 879	. . 3 ⊢ Ⅎ𝑦(𝜑 ↔ [𝑥 / 𝑦][𝑦 / 𝑥]𝜑)
4		ndpsub2-P7.14 839	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
5		eqsym-P7.CL.SYM 938	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
6	4, 5	subiml2-P4.RC 541	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
7		ndpsub2-P7.14 839	. . . . 5 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦][𝑦 / 𝑥]𝜑))
8	6, 7	bitrns-P3.33c 302	. . . 4 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦][𝑦 / 𝑥]𝜑))
9		lemma-L7.01a 924	. . . . 5 ⊢ ([𝑦 / 𝑧] 𝑧 = 𝑥 ↔ 𝑦 = 𝑥)
10	9	bisym-P3.33b.RC 299	. . . 4 ⊢ (𝑦 = 𝑥 ↔ [𝑦 / 𝑧] 𝑧 = 𝑥)
11	8, 10	subiml2-P4.RC 541	. . 3 ⊢ ([𝑦 / 𝑧] 𝑧 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦][𝑦 / 𝑥]𝜑))
12		axL6ex-P7 925	. . 3 ⊢ ∃𝑧 𝑧 = 𝑥
13	3, 11, 12	ndexew-P7.RC.VR1of2 888	. 2 ⊢ (𝜑 ↔ [𝑥 / 𝑦][𝑦 / 𝑥]𝜑)
14	13	bisym-P3.33b.RC 299	1 ⊢ ([𝑥 / 𝑦][𝑦 / 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:  psubid-P7  940