Theorem bnj118 35566

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj118.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj118.2	⊢ (𝜑′ ↔ [1o / 𝑛]𝜑)

Assertion

Ref	Expression
bnj118	⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))

Distinct variable groups:   𝐴,𝑛   𝑅,𝑛   𝑓,𝑛   𝑥,𝑛

Allowed substitution hints:   𝜑(𝑥, 𝑓, 𝑛)   𝐴(𝑥, 𝑓)   𝑅(𝑥, 𝑓)   𝜑′(𝑥, 𝑓, 𝑛)

Proof of Theorem bnj118

Step	Hyp	Ref	Expression
1		bnj118.2	. 2 ⊢ (𝜑′ ↔ [1o / 𝑛]𝜑)
2		bnj118.1	. . 3 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
3		bnj105 35422	. . 3 ⊢ 1o ∈ V
4	2, 3	bnj91 35558	. 2 ⊢ ([1o / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
5	1, 4	bitri 278	1 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  [wsbc 3742  ∅c0 4281  ‘cfv 6555  1_oc1o 8494   predc-bnj14 35386

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-pw 4562  df-sn 4589  df-suc 6383  df-1o 8501

This theorem is used by:  bnj151  35574  bnj153  35577