Theorem psubvar1-P6 802

Description: Proper Substitution Applied to Atomic Term (same variable).

'𝑎' is distinct from all other variables.

Assertion

Ref	Expression
psubvar1-P6	⊢ ([𝑡 / 𝑥] 𝑎 = 𝑥 ↔ 𝑎 = 𝑡)

Distinct variable groups:   𝑡,𝑎   𝑥,𝑎

Proof of Theorem psubvar1-P6

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfrv-P6 686	. 2 ⊢ Ⅎ𝑥 𝑎 = 𝑦
2		nfrv-P6 686	. 2 ⊢ Ⅎ𝑦 𝑎 = 𝑡
3		subeqr-P5.CL 636	. 2 ⊢ (𝑥 = 𝑦 → (𝑎 = 𝑥 ↔ 𝑎 = 𝑦))
4		subeqr-P5.CL 636	. 2 ⊢ (𝑦 = 𝑡 → (𝑎 = 𝑦 ↔ 𝑎 = 𝑡))
5	1, 2, 3, 4	isubtopsub-P6 729	1 ⊢ ([𝑡 / 𝑥] 𝑎 = 𝑥 ↔ 𝑎 = 𝑡)

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: (None)