Theorem lemma-L7.02a 944

Description: Proper Substitution Over Implication Lemma. †

Hypotheses

Ref	Expression
lemma-L7.02a.1	⊢ Ⅎ𝑥𝛾
lemma-L7.02a.2	⊢ (𝛾 → (𝜑 → 𝜓))

Assertion

Ref	Expression
lemma-L7.02a	⊢ (𝛾 → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))

Proof of Theorem lemma-L7.02a

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ndpsub4-P7.16 841	. . . 4 ⊢ ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑦][𝑦 / 𝑥]𝜑)
2	1	rcp-NDBIEF0 240	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑦][𝑦 / 𝑥]𝜑)
3	2	rcp-NDIMP0addall 207	. 2 ⊢ (𝛾 → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑦][𝑦 / 𝑥]𝜑))
4		ndnfrv-P7.1 826	. . 3 ⊢ Ⅎ𝑦𝛾
5		lemma-L7.02a.1	. . . 4 ⊢ Ⅎ𝑥𝛾
6		lemma-L7.02a.2	. . . 4 ⊢ (𝛾 → (𝜑 → 𝜓))
7	5, 6	lemma-L7.02a-L1 943	. . 3 ⊢ (𝛾 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
8	4, 7	lemma-L7.02a-L1 943	. 2 ⊢ (𝛾 → ([𝑡 / 𝑦][𝑦 / 𝑥]𝜑 → [𝑡 / 𝑦][𝑦 / 𝑥]𝜓))
9		ndpsub4-P7.16 841	. . . 4 ⊢ ([𝑡 / 𝑥]𝜓 ↔ [𝑡 / 𝑦][𝑦 / 𝑥]𝜓)
10	9	rcp-NDBIER0 241	. . 3 ⊢ ([𝑡 / 𝑦][𝑦 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜓)
11	10	rcp-NDIMP0addall 207	. 2 ⊢ (𝛾 → ([𝑡 / 𝑦][𝑦 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜓))
12	3, 8, 11	dsyl-P3.25 261	1 ⊢ (𝛾 → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))

Syntax hints:  term-obj 1   → wff-imp 10  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-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:  axL4-P7  945  axL4ex-P7  946