Theorem ndpsub1-P7.13.VR 863

Description: ndpsub1-P7.13 838 with extra variable restriction. †

'𝑥' cannot occur in either '𝑡' or '𝜓'.

Hypothesis

Ref	Expression
ndpsub1-P7.13.VR.1	⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓))

Assertion

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

Distinct variable groups:   𝜓,𝑥   𝑡,𝑥

Proof of Theorem ndpsub1-P7.13.VR

Step	Hyp	Ref	Expression
1		ndnfrv-P7.1 826	. 2 ⊢ Ⅎ𝑥𝜓
2		ndpsub1-P7.13.VR.1	. 2 ⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓))
3	1, 2	ndpsub1-P7.13 838	1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ 𝜓)

Syntax hints:  term-obj 1   = wff-equals 6   → wff-imp 10   ↔ 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:  lemma-L7.01a  924