Theorem ndpsub1-P7.13 838

Description: Natural Deduction: Proper Substitution Rule 1.

'𝑥' cannot occur in '𝑡'.

Hypotheses

Ref	Expression
ndpsub1-P7.13.1	⊢ Ⅎ𝑥𝜓
ndpsub1-P7.13.2	⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ndpsub1-P7.13	⊢ ([𝑡 / 𝑥]𝜑 ↔ 𝜓)

Distinct variable group:   𝑡,𝑥

Proof of Theorem ndpsub1-P7.13

Step	Hyp	Ref	Expression
1		ndpsub1-P7.13.1	. 2 ⊢ Ⅎ𝑥𝜓
2		ndpsub1-P7.13.2	. 2 ⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓))
3	1, 2	isubtopsubv-P6 727	1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ 𝜓)

Syntax hints:  term-obj 1   = wff-equals 6   → wff-imp 10   ↔ 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-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:  ndpsub1-P7.13.VR  863  cbvall-P7-L1  1060  cbvex-P7-L1  1065