Theorem psubid-P7 940

Description: Proper Substitution Identity Property. †

Assertion

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

Proof of Theorem psubid-P7

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ndpsub4-P7.16 841	. 2 ⊢ ([𝑥 / 𝑥]𝜑 ↔ [𝑥 / 𝑦][𝑦 / 𝑥]𝜑)
2		ndnfrv-P7.1 826	. . 3 ⊢ Ⅎ𝑦𝜑
3	2	psubinv-P7 939	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ 𝜑)
4	1, 3	bitrns-P3.33c.RC 303	1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:  term-obj 1   ↔ wff-bi 104  [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:  alle-P7  941  exi-P7  951