Theorem subnfr-P6 755

Description: Substitution Law for 'Ⅎ𝑥' Predicate.

Hypotheses

Ref	Expression
subnfr.1	⊢ Ⅎ𝑥𝛾
subnfr.2	⊢ (𝛾 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
subnfr-P6	⊢ (𝛾 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))

Proof of Theorem subnfr-P6

Step	Hyp	Ref	Expression
1		subnfr.1	. . . 4 ⊢ Ⅎ𝑥𝛾
2		subnfr.2	. . . 4 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
3	1, 2	subex-P6 754	. . 3 ⊢ (𝛾 → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
4	1, 2	suball-P6 753	. . 3 ⊢ (𝛾 → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
5	3, 4	subimd-P3.40c 329	. 2 ⊢ (𝛾 → ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ (∃𝑥𝜓 → ∀𝑥𝜓)))
6		dfnfreealt-P6 683	. . . 4 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
7	6	bisym-P3.33b.RC 299	. . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ Ⅎ𝑥𝜑)
8	7	rcp-NDIMP0addall 207	. 2 ⊢ (𝛾 → ((∃𝑥𝜑 → ∀𝑥𝜑) ↔ Ⅎ𝑥𝜑))
9		dfnfreealt-P6 683	. . . 4 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
10	9	bisym-P3.33b.RC 299	. . 3 ⊢ ((∃𝑥𝜓 → ∀𝑥𝜓) ↔ Ⅎ𝑥𝜓)
11	10	rcp-NDIMP0addall 207	. 2 ⊢ (𝛾 → ((∃𝑥𝜓 → ∀𝑥𝜓) ↔ Ⅎ𝑥𝜓))
12	5, 8, 11	subbid2-P4 550	1 ⊢ (𝛾 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))

Syntax hints:  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104  ∃wff-exists 595  Ⅎwff-nfree 681

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

This theorem is referenced by:  subnfr-P6.VR  756  nfrsucc-P6  780  nfradd-P6  781  nfrmult-P6  782  ndnfrleq-P7.11  836