Theorem gencbval 3550

Description: Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.)

Hypotheses

Ref	Expression
gencbval.1	⊢ 𝐴 ∈ V
gencbval.2	⊢ (𝐴 = 𝑦 → (𝜑 ↔ 𝜓))
gencbval.3	⊢ (𝐴 = 𝑦 → (𝜒 ↔ 𝜃))
gencbval.4	⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝑦))

Assertion

Ref	Expression
gencbval	⊢ (∀𝑥(𝜒 → 𝜑) ↔ ∀𝑦(𝜃 → 𝜓))

Distinct variable groups:   𝜓,𝑥   𝜑,𝑦   𝜃,𝑥   𝜒,𝑦   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑥)   𝜃(𝑦)   𝐴(𝑥)

Proof of Theorem gencbval

Step	Hyp	Ref	Expression
1		gencbval.1	. . . 4 ⊢ 𝐴 ∈ V
2		gencbval.2	. . . . 5 ⊢ (𝐴 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	notbid 320	. . . 4 ⊢ (𝐴 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
4		gencbval.3	. . . 4 ⊢ (𝐴 = 𝑦 → (𝜒 ↔ 𝜃))
5		gencbval.4	. . . 4 ⊢ (𝜃 ↔ ∃𝑥(𝜒 ∧ 𝐴 = 𝑦))
6	1, 3, 4, 5	gencbvex 3548	. . 3 ⊢ (∃𝑥(𝜒 ∧ ¬ 𝜑) ↔ ∃𝑦(𝜃 ∧ ¬ 𝜓))
7		exanali 1853	. . 3 ⊢ (∃𝑥(𝜒 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝜒 → 𝜑))
8		exanali 1853	. . 3 ⊢ (∃𝑦(𝜃 ∧ ¬ 𝜓) ↔ ¬ ∀𝑦(𝜃 → 𝜓))
9	6, 7, 8	3bitr3i 303	. 2 ⊢ (¬ ∀𝑥(𝜒 → 𝜑) ↔ ¬ ∀𝑦(𝜃 → 𝜓))
10	9	con4bii 323	1 ⊢ (∀𝑥(𝜒 → 𝜑) ↔ ∀𝑦(𝜃 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1529   = wceq 1531  ∃wex 1774   ∈ wcel 2108  Vcvv 3493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-11 2154  ax-12 2170  ax-ext 2791

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1775  df-nf 1779  df-cleq 2812  df-clel 2891

This theorem is referenced by: (None)