Theorem lemma-L7.01a 924

Description: Proper Substitution Over Equality Lemma. †

'𝑥' cannot occur in '𝑡' or '𝑢'.

Assertion

Ref	Expression
lemma-L7.01a	⊢ ([𝑡 / 𝑥] 𝑥 = 𝑢 ↔ 𝑡 = 𝑢)

Distinct variable groups:   𝑡,𝑥   𝑢,𝑥

Proof of Theorem lemma-L7.01a

Step	Hyp	Ref	Expression
1		ndsubeql-P7.22a.CL 911	. 2 ⊢ (𝑥 = 𝑡 → (𝑥 = 𝑢 ↔ 𝑡 = 𝑢))
2	1	ndpsub1-P7.13.VR 863	1 ⊢ ([𝑡 / 𝑥] 𝑥 = 𝑢 ↔ 𝑡 = 𝑢)

Syntax hints:  term-obj 1   = wff-equals 6   ↔ 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-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:  axL6ex-P7  925  psubnfrv-P7  927  psubinv-P7  939  lemma-L7.02a-L1  943  lemma-L7.03  962