Theorem solvesub-P6b 707

Description: solvesub-P6a 704 with result substituted back into hypothesis.

Requires the existence of '𝜓₁(𝑥₁)' as a replacement for '𝜓(𝑥)'. Also, '𝑥' cannot occur in '𝑡'.

Hypotheses

Ref	Expression
solvesub-P6b.1	⊢ (𝑥 = 𝑥₁ → (𝜓 ↔ 𝜓₁))
solvesub-P6b.2	⊢ Ⅎ𝑥𝜓
solvesub-P6b.3	⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
solvesub-P6b	⊢ (𝑥 = 𝑡 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))

Distinct variable groups:   𝜓,𝑥₁   𝜓₁,𝑥   𝑡,𝑥   𝑥,𝑥₁

Proof of Theorem solvesub-P6b

Step	Hyp	Ref	Expression
1		solvesub-P6b.3	. 2 ⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓))
2		solvesub-P6b.1	. . . 4 ⊢ (𝑥 = 𝑥₁ → (𝜓 ↔ 𝜓₁))
3		solvesub-P6b.2	. . . 4 ⊢ Ⅎ𝑥𝜓
4	2, 3, 1	solvesub-P6a 704	. . 3 ⊢ (𝜓 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
5	4	subbir-P3.41b.RC 335	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))
6	1, 5	subimr2-P4.RC 543	1 ⊢ (𝑥 = 𝑡 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)))

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104  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

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: (None)