Theorem wfis2fgOLD 6378

Description: Obsolete version of wfis2fg 6377 as of 17-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 11-Feb-2011.)

Hypotheses

Ref	Expression
wfis2fgOLD.1	⊢ Ⅎ𝑦𝜓
wfis2fgOLD.2	⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))
wfis2fgOLD.3	⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑))

Assertion

Ref	Expression
wfis2fgOLD	⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑦,𝐴,𝑧   𝜑,𝑧   𝑦,𝑅,𝑧

Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑦,𝑧)

Proof of Theorem wfis2fgOLD

Step	Hyp	Ref	Expression
1		sbsbc 3792	. . . . 5 ⊢ ([𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑)
2		wfis2fgOLD.1	. . . . . 6 ⊢ Ⅎ𝑦𝜓
3		wfis2fgOLD.2	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))
4	2, 3	sbiev 2314	. . . . 5 ⊢ ([𝑧 / 𝑦]𝜑 ↔ 𝜓)
5	1, 4	bitr3i 277	. . . 4 ⊢ ([𝑧 / 𝑦]𝜑 ↔ 𝜓)
6	5	ralbii 3093	. . 3 ⊢ (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓)
7		wfis2fgOLD.3	. . 3 ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑))
8	6, 7	biimtrid 242	. 2 ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑))
9	8	wfisg 6374	1 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  Ⅎwnf 1783  [wsb 2064   ∈ wcel 2108  ∀wral 3061  [wsbc 3788   Se wse 5635   We wwe 5636  Predcpred 6320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321

This theorem is referenced by: (None)